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

0,4

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).

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).

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 *)

Prog

(PARI) {a(n) = sum(k=0, floor(n/2), binomial(3*k, n-2*k))}; \\ G. C. Greubel, May 09 2019
(Magma) [(&+[Binomial(3*k, n-2*k): k in [0..Floor(n/2)]]): n in [0..40]]; // 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

Crossrefs

Sequence in context: A130755 A361635 A286348 * A287741 A291710 A100455

Adjacent sequences: A116087 A116088 A116089 * A116091 A116092 A116093

Keyword

easy,nonn

Author

Paul Barry, Feb 04 2006

Status

approved