Search: id:A001792 Results 1-1 of 1 results found. %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, Table of n, a(n) for n=0..500 %H A001792 Milan Janjic, Two Enumerative Functions %H A001792 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. %H A001792 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. %H A001792 F. Ellermann, Illustration of binomial transforms %H A001792 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 146 %H A001792 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4. %H A001792 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A001792 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A001792 N. J. A. Sloane, Transforms %H A001792 Index entries for sequences related to linear recurrences with constant coefficients %H A001792 Index entries for sequences related to Chebyshev polynomials. %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 (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 Search completed in 0.003 seconds