A116090 Expansion of 1/(1-x^2*(1+x)^3).

1, 0, 1, 3, 4, 7, 16, 29, 52, 102, 194, 361, 685, 1301, 2452, 4633, 8771, 16577, 31327, 59241, 112004, 211724, 400285, 756786, 1430710, 2704817, 5113647, 9667590, 18277014, 34553692, 65325542, 123501151, 233485250, 441415867, 834519021

Offset: 0

Author

Paul Barry, Feb 04 2006

Keywords

• easy
• nonn

Comments

Diagonal sums of number triangle A116089.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0, 1, 3, 3, 1).

Programs

Magma

[(&+[Binomial(3*k, n-2*k): k in [0..Floor(n/2)]]): n in [0..40]]; // G. C. Greubel, May 09 2019

Mathematica

CoefficientList[Series[1/(1-x^2(1+x)^3),{x,0,40}],x] (* or *) LinearRecurrence[{0,1,3,3,1},{1,0,1,3,4},40] (* Harvey P. Dale, Apr 28 2014 *)

PARI

{a(n) = sum(k=0, floor(n/2), binomial(3*k, n-2*k))}; \\ G. C. Greubel, May 09 2019

Sage

[sum(binomial(3*k, n-2*k) for k in (0..floor(n/2))) for n in (0..40)] # G. C. Greubel, May 09 2019

Formula

a(n) = a(n-2) + 3*a(n-3) + 3*a(n-4) + a(n-5).

a(n) = Sum_{k=0..floor(n/2)} C(3*k, n-2*k).

a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*C(4*k,n-k)/C(4*k,k).