A160906 Row sums of A159841.

1, 5, 29, 176, 1093, 6885, 43796, 280600, 1807781, 11698223, 75973189, 494889092, 3231947596, 21153123932, 138712176296, 911137377456, 5993760282021, 39481335979779, 260377117268087, 1719026098532296, 11360252318843933, 75141910203168229, 497431016774189912

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=0..n} A159841(n,k).

Conjecture: a(2n+1) = A075273(3n).

a(n) = C(3*n+1,n)*Hyper2F1([1,-n],[2*n+2],-1). - Peter Luschny, May 19 2015

Conjecture: 2*n*(2*n-1)*(5*n-4)*a(n) +(-295*n^3+451*n^2-130*n-24)*a(n-1) +24*(5*n+1)*(3*n-4)*(3*n-2)*a(n-2) = 0. - R. J. Mathar, Jul 20 2016

R. J. Mathar's conjecture verified using differential equation (2112*x + 2)*y + (11328*x^2 - 842*x + 2)*y' + (7776*x^3 - 1319*x^2 + 34*x)*y'' + (1080*x^4 - 295*x^3 + 20*x^2)*y''' - 12 satisfied by the g.f. y(x). - Robert Israel, Mar 13 2026

a(n) = [x^n] 1/((1 - 2*x)*(1 - x)^(2*n+1)). - Ilya Gutkovskiy, Oct 25 2017

a(n) ~ 3^(3*n + 3/2) / (sqrt(Pi*n) * 2^(2*n + 1)). - Vaclav Kotesovec, Oct 25 2017

a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+k,k). - Seiichi Manyama, Aug 03 2025

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k). - Seiichi Manyama, Aug 07 2025

G.f.: g^2/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 12 2025

G.f.: B(x)^2/(1 + (B(x)-1)/3), where B(x) is the g.f. of A005809. - Seiichi Manyama, Aug 15 2025

G.f.: 1/(1 - x*g*(6-g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 16 2025

Maple

A160906 := proc(n) add( A159841(n, k), k=0..n) ; end:
seq(A160906(n), n=0..20) ;

Mathematica

Table[Sum[Binomial[3*n+1, 2*n+k+1], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 25 2017 *)

Prog

(SageMath)
a = lambda n: binomial(3*n+1, n)*hypergeometric([1, -n], [2*n+2], -1)
[simplify(a(n)) for n in range(21)] # Peter Luschny, May 19 2015
(PARI) a(n) = sum(k=0, n, binomial(3*n+1, 2*n+k+1)); \\ Michel Marcus, Oct 31 2017

Crossrefs

Cf. A075273, A159841.

Cf. A005809, A006256, A183160, A226751.

Cf. A000302, A386811, A386812.

Cf. A244038, A383326.

Sequence in context: A384368 A390588 A163611 * A163073 A190802 A139174

Adjacent sequences: A160903 A160904 A160905 * A160907 A160908 A160909

Keyword

nonn,easy

Author

R. J. Mathar, May 29 2009

Status

approved