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
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Crossrefs
Programs
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Maple
A160906 := proc(n) add( A159841(n,k), k=0..n) ; end: seq(A160906(n), n=0..20) ;
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Mathematica
Table[Sum[Binomial[3*n+1, 2*n+k+1], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 25 2017 *) -
PARI
a(n) = sum(k=0, n, binomial(3*n+1, 2*n+k+1)); \\ Michel Marcus, Oct 31 2017
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Sage
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
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
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