0,2

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

Binomial transform of natural numbers [1,2,3,4,...].

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

The arithmetic mean of first n terms of the sequence is 2^n. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Dec 25 2001

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

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

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

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

a(n)=A018804(2^n). - Matthew Vandermast (ghodges14(AT)comcast.net), Mar 01 2003

a(n)=(1/4)A001787(n+2) - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 24 2003

Number of ones in all different (n+1)-bit integers. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Aug 02 2003

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

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

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

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

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]

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]

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]

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]

a(n) = A164910(2^n). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 30 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

M. Hirschhorn, Calkin's binomial identity, Discr. Math., 159 (1996), 273-278.

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.

Maohua Le, Two Classes of Smarandache Determinants, Scientia Magna, vol 2, no 1 (2006), pp 20-25.

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).

Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.

T. D. Noe, Table of n, a(n) for n=0..500

Milan Janjic, Two Enumerative Functions

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].

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

F. Ellermann, Illustration of binomial transforms

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 146

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

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.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

N. J. A. Sloane, Transforms

Index entries for sequences related to linear recurrences with constant coefficients

Index entries for sequences related to Chebyshev polynomials.

G.f.: (1-x)/(1-2*x)^2. a(n)=4*a(n-1)-4*a(n-2).

a(n) = Sum{k=0..(n+2), binomial(n+2, 2k)*k } - Paul Barry (pbarry(AT)wit.ie), Mar 06 2003

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

a(n)=sum_(k=0, ..., n) binomial(n, k)*(k+1). - Lekraj Beedassy(AT)hotmail.com (blekraj(AT)yahoo.com), Jun 24 2004

a(n) = A000244(n) - A066810(n). - Ross La Haye (rlahaye(AT)new.rr.com), Apr 29 2006

Row sums of triangle A130585 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 06 2007

Equals A125092 * [1/1, 1/2, 1/3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 16 2007

a(n) = (n+1)*2^n - n*2^(n-1). Equals A128064 * A000079 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 28 2007

G.f.: F(3, 1; 2; 2x); [From Paul Barry (pbarry(AT)wit.ie), Sep 03 2008]

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]

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]

a(n)=2*a(n-1)+2^(n-1). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 19 2009]

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]

A001792 := n-> (n+2)*2^(n-1);

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

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

A001792:=-(-3+4*z)/(2*z-1)^2; [S. Plouffe in his 1992 dissertation. Gives sequence without the initial 1.]

seq(add(iquo(2^n, 2), k=0..n)/2, n=1..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 20 2008

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]

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)

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]

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)

abs(det(toeplitz(1:n))) - Paul M. Payton (paul.payton(AT)lmco.com), May 23 2006

(PARI) A001792(n)=(n+2)<<(n-1) [From M. F. Hasler (MHasler(AT)univ-ag.fr), Dec 17 2008]

First differences of A001787. a(n)=A049600(n, 1), a(n)= A030523(n+1, 1). Cf. A053113.

Row sums of triangles A008949 and A055248. a(n)= -A039991(n+2, 2).

Cf. A001787, A130584, A125092, A128064, A000079.

A164910 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 30 2009]

Sequence in context: A050233 A143662 A049610 this_sequence A018795 A018794 A018793

Adjacent sequences: A001789 A001790 A001791 this_sequence A001793 A001794 A001795

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Ralf Stephan (ralf(AT)ark.in-berlin.de), Aug 02 2003

Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009