%I A001792 M2739 N1100
%S A001792 1,3,8,20,48,112,256,576,1280,2816,6144,13312,28672,61440,131072,
%T A001792 278528,589824,1245184,2621440,5505024,11534336,24117248,50331648,
%U A001792 104857600,218103808,452984832,939524096,1946157056,4026531840
%N A001792 (n+2)*2^(n-1).
%C A001792 Number of parts in all compositions (ordered partitions) of n+1. For 
               example, a(2)=8 because in 3=2+1=1+2=1+1+1 we have 8 parts. Also 
               number of compositions (ordered partitions) of 2n+1 with exactly 
               1 odd part. For example, a(2)=8 because the only compositions of 
               5 with exactly 1 odd part are = 5=1+4=2+3=3+2=4+1=1+2+2=2+1+2=2+2+1. 
               - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 10 2001
%C A001792 Binomial transform of natural numbers [1,2,3,4,...].
%C A001792 For n >= 1 a(n) is also the determinant of the n X n matrix with 3's 
               on the diagonal and 1's elsewhere. - Ahmed Fares (ahmedfares(AT)my-deja.com), 
               May 06 2001
%C A001792 The arithmetic mean of first n terms of the sequence is 2^n. - Amarnath 
               Murthy (amarnath_murthy(AT)yahoo.com), Dec 25 2001
%C A001792 Also the number of "winning paths" of length n across an n X n Hex board. 
               Satisfies the recursion a(n)=2a(n-1)+2^{n-2}. - David Molnar (molnar(AT)stolaf.edu), 
               Apr 10 2002
%C A001792 Let m(i,1)=i; m(1,j)=j; m(i,j)=m(i,j-1)+m(i-1,j-1); then a(n)=m(n,n) 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), May 08 2002
%C A001792 Let M_n be the n X n matrix m_(i,j)=1+abs(i-j) then det(M_n)=(-1)^(n-1)*a(n-1) 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), May 28 2002
%C A001792 Absolute value of determinant of n X n matrix of form : [1 2 3 4 5 / 
               2 1 2 3 4 / 3 2 1 2 3 / 4 3 2 1 2 / 5 4 3 2 1] - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jun 20 2002
%C A001792 a(n)=A018804(2^n). - Matthew Vandermast (ghodges14(AT)comcast.net), Mar 
               01 2003
%C A001792 a(n)=(1/4)A001787(n+2) - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 
               24 2003
%C A001792 Number of ones in all different (n+1)-bit integers. - Ralf Stephan (ralf(AT)ark.in-berlin.de), 
               Aug 02 2003
%C A001792 This sequence also emerges as a floretion force transform of powers of 
               2 (see program code). Define a(-1) = 0 (as the sequence is returned 
               by FAMP). Then a(n-1) + A098156(n+1) = 2*a(n) (conjecture) - Creighton 
               Dement (creighton.k.dement(AT)uni-oldenburg.de), Mar 14 2005
%C A001792 This sequence gives the absolute value of the determinant of the Toeplitz 
               matrix with first row containing the first n integers. - Paul M. 
               Payton (paul.payton(AT)lmco.com), May 23 2006
%C A001792 Equals sums of rows right of left edge of A102363 divided by three, + 
               2^K - David G. Williams (davidwilliams(AT)paxway.com), Oct 08 2007
%C A001792 If X_1,X_2,...,X_n are 2-blocks of a (2n+1)-set X then, for n>=1, a(n) 
               is the number of (n+1)-subsets of X intersecting each X_i, (i=1,2,
               ...,n). - Milan R. Janjic (agnus(AT)blic.net), Nov 18 2007
%C A001792 Also, a(n-1) is the determinant of the n x n matrix with A[i,j] = n-|i-j|. 
               [From M. F. Hasler (MHasler(AT)univ-ag.fr), Dec 17 2008]
%C A001792 1/2 the number of permutations of 1..(n+2) arranged in a circle with 
               exactly one local maximum. [From Ron Hardin (rhhardin(AT)att.net), 
               Apr 19 2009]
%C A001792 The first corrector line for transforming 2^n offset 0 with a leading 
               1 into the fibonacci sequence. [From Al Hakanson (hawkuu(AT)gmail.com), 
               Jun 01 2009]
%C A001792 a(n) is the number of runs of consecutive 1's in all binary sequences 
               of length (n+1). [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), 
               Jul 02 2009]
%C A001792 a(n) = A164910(2^n). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 
               30 2009]
%D A001792 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001792 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001792 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 795.
%D A001792 M. Hirschhorn, Calkin's binomial identity, Discr. Math., 159 (1996), 
               273-278.
%D A001792 Jones, C. W.; Miller, J. C. P.; Conn, J. F. C.; Pankhurst, R. C.; Tables 
               of Chebyshev polynomials. Proc. Roy. Soc. Edinburgh. Sect. A. 62, 
               (1946). 187-203.
%D A001792 Maohua Le, Two Classes of Smarandache Determinants, Scientia Magna, vol 
               2, no 1 (2006), pp 20-25.
%D A001792 A. M. Stepin and A. T. Tagi-Zade, Words with restrictions, pp. 67-74 
               of Qvant Selecta: Combinatorics I, Amer. Math. Soc., 2001 (G_n on 
               p. 70).
%D A001792 Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, 
               Discrete Math., 274 (2004), 331-342.
%H A001792 T. D. Noe, <a href="b001792.txt">Table of n, a(n) for n=0..500</a>
%H A001792 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A001792 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A001792 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A001792 F. Ellermann, <a href="a001792.txt">Illustration of binomial transforms</
               a>
%H A001792 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=146">
               Encyclopedia of Combinatorial Structures 146</a>
%H A001792 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               On generalizations of Stirling number triangles</a>, J. Integer Seqs., 
               Vol. 3 (2000), #00.2.4.
%H A001792 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001792 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001792 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A001792 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A001792 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A001792 G.f.: (1-x)/(1-2*x)^2. a(n)=4*a(n-1)-4*a(n-2).
%F A001792 a(n) = Sum{k=0..(n+2), binomial(n+2, 2k)*k } - Paul Barry (pbarry(AT)wit.ie), 
               Mar 06 2003
%F A001792 With a leading 0, this is ((n+1)2^n-0^n)/4=sum{m=0..n, C(n-1, m-1)m }, 
               the binomial transform of A004526(n+1). - Paul Barry (pbarry(AT)wit.ie), 
               Jun 05 2003
%F A001792 a(n)=sum_(k=0, ..., n) binomial(n, k)*(k+1). - Lekraj Beedassy(AT)hotmail.com 
               (blekraj(AT)yahoo.com), Jun 24 2004
%F A001792 a(n) = A000244(n) - A066810(n). - Ross La Haye (rlahaye(AT)new.rr.com), 
               Apr 29 2006
%F A001792 Row sums of triangle A130585 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jun 06 2007
%F A001792 Equals A125092 * [1/1, 1/2, 1/3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Nov 16 2007
%F A001792 a(n) = (n+1)*2^n - n*2^(n-1). Equals A128064 * A000079 - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Dec 28 2007
%F A001792 G.f.: F(3, 1; 2; 2x); [From Paul Barry (pbarry(AT)wit.ie), Sep 03 2008]
%F A001792 a(n)=1+sum{k=1...n,(n-k+4)2^(n-k-1)} This follows from the result that 
               the number of parts equal to k in all compostions of n is (n-k+3)2^(n-k-2) 
               for 0<k<n. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), 
               Sep 21 2008]
%F A001792 a(n) = 2^(n-1) + 2 a(n-1) ; a(n-1) = det(n-|i-j|)_{i,j=1...n} [From M. 
               F. Hasler (MHasler(AT)univ-ag.fr), Dec 17 2008]
%F A001792 a(n)=2*a(n-1)+2^(n-1). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Apr 19 2009]
%e A001792 a(0)=1, a(1)=2*1+1=3, a(2)=2*3+2=8, a(3)=2*8+4=20, a(4)=2*20+8=48, a(5)=2*48+16=112, 
               a(6)=2*112+32=256,... [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Apr 19 2009]
%p A001792 A001792 := n-> (n+2)*2^(n-1);
%p A001792 spec := [S, {B=Set(Z, 0 <= card), S=Prod(Z, B, B)}, labeled]: seq(combstruct[count](spec, 
               size=n)/4, n=2..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Oct 09 2006
%p A001792 a:=n->1/8*sum(sum (2*binomial(n,k),j=0..n),k=0..n): seq(a(n),n=1..29); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 05 2007
%p A001792 A001792:=-(-3+4*z)/(2*z-1)^2; [S. Plouffe in his 1992 dissertation. Gives 
               sequence without the initial 1.]
%p A001792 seq(add(iquo(2^n,2),k=0..n)/2,n=1..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 20 2008
%p A001792 restart: G(x):=1/exp(2*x)*(1-x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],
               x) od: x:=0: seq(abs(f[n]),n=0..28 );# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 17 2009]
%t A001792 matrix[n_Integer /; n >= 1] := Table[Abs[p - q] + 1, {q, n}, {p, n}]; 
               a[n_Integer /; n >= 1] := Abs[Det[matrix[n]]] (from Josh Locker (joshlocker(AT)macfora.com), 
               Apr 29 2004)
%t A001792 g[n_,m_,r_]:=Binomial[n-1,r-1] Binomial[m+1,r] r; Table[1 + Sum[g[n, 
               k - n, r], {r, 1, k}, {n, 1, k - 1}], {k, 1, 29}] [From Geoffrey 
               Critzer (critzer.geoffrey(AT)usd443.org), Jul 02 2009]
%o A001792 Floretion Algebra Multiplication Program, FAMP Code: 2jesforseq[ - .25'i 
               - .25'j - .25'kk' - .25'ij' - .25'ik' - .25'ji' + .25'jk' + .25e], 
               1vesforseq(n) = 2^n, ForType: 1A. Identity used: jesleft + jesright 
               = jes. (Dement)
%o A001792 abs(det(toeplitz(1:n))) - Paul M. Payton (paul.payton(AT)lmco.com), May 
               23 2006
%o A001792 (PARI) A001792(n)=(n+2)<<(n-1) [From M. F. Hasler (MHasler(AT)univ-ag.fr), 
               Dec 17 2008]
%Y A001792 First differences of A001787. a(n)=A049600(n, 1), a(n)= A030523(n+1, 
               1). Cf. A053113.
%Y A001792 Row sums of triangles A008949 and A055248. a(n)= -A039991(n+2, 2).
%Y A001792 Cf. A001787, A130584, A125092, A128064, A000079.
%Y A001792 A164910 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 30 2009]
%Y A001792 Sequence in context: A050233 A143662 A049610 this_sequence A018795 A018794 
               A018793
%Y A001792 Adjacent sequences: A001789 A001790 A001791 this_sequence A001793 A001794 
               A001795
%K A001792 nonn,easy,nice
%O A001792 0,2
%A A001792 N. J. A. Sloane (njas(AT)research.att.com).
%E A001792 More terms from Ralf Stephan (ralf(AT)ark.in-berlin.de), Aug 02 2003
%E A001792 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 11 2009