A005726 Quadrinomial coefficients. (Formerly M1643)

1, 2, 6, 20, 65, 216, 728, 2472, 8451, 29050, 100298, 347568, 1208220, 4211312, 14712960, 51507280, 180642391, 634551606, 2232223626, 7862669700, 27727507521, 97884558992, 345891702456, 1223358393120, 4330360551700

Offset

1,2

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

R. K. Guy, Letter to N. J. A. Sloane, 1987

Formula

a(n) = Sum_{k=0..floor(n/2)}, C(n,k) C(n,2k+1). - Paul Barry, May 15 2003

a(n) = Sum[(-1)^k binomial[n,k] binomial[2n-2-4k,n-1],{k,0,Floor[(n-1)/4]}]. - David Callan, Jul 03 2006

G.f.: F(G^(-1)(x)) where F(t) = (t-1)^2*(t^2+1)^2/(2*t^3-t^2+1) and G(t) = t/((t-1)*(t^2+1)). - Mark van Hoeij, Oct 30 2011

Conjecture: 2*(n-1)*(2*n+1)*(13*n-14)*a(n) +(-143*n^3+297*n^2-148*n+12) *a(n-1) -4*(n-1)*(26*n^2-41*n+9)*a(n-2) -16*(n-1)*(n-2)*(13*n-1) *a(n-3)=0. - R. J. Mathar, Nov 13 2012

a(n) = A008287(n,n-1). - Sean A. Irvine, Aug 15 2016

Maple

for n from 1 to 40 do printf(`%d, `, coeff(expand(sum(x^j, j=0..3)^n), x, n-1)) od:
F := (t-1)^2*(t^2+1)^2/(2*t^3-t^2+1);  G := t/((t-1)*(t^2+1)); Ginv := RootOf(numer(G-x), t);  ogf := series(eval(F, t=Ginv), x=0, 20); # Mark van Hoeij, Oct 30 2011

Mathematica

Table[Sum[Binomial[n, k]Binomial[n, 2k+1], {k, 0, Floor[n/2]}], {n, 30}] (* Harvey P. Dale, Oct 19 2013 *)

Crossrefs

Sequence in context: A273902 A181301 A302612 * A148473 A000718 A148474

Adjacent sequences: A005723 A005724 A005725 * A005727 A005728 A005729

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Aug 21 2000

Status

approved