A005726 Quadrinomial coefficients.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Programs

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 *)

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

Extensions

More terms from James Sellers, Aug 21 2000