cp's OEIS Frontend

A160906 Row sums of A159841.

%I A160906 #43 Aug 16 2025 10:25:14
%S A160906 1,5,29,176,1093,6885,43796,280600,1807781,11698223,75973189,
%T A160906 494889092,3231947596,21153123932,138712176296,911137377456,
%U A160906 5993760282021,39481335979779,260377117268087,1719026098532296,11360252318843933,75141910203168229,497431016774189912
%N A160906 Row sums of A159841.
%H A160906 Vincenzo Librandi, <a href="/A160906/b160906.txt">Table of n, a(n) for n = 0..1000</a>
%F A160906 a(n) = Sum_{k=0..n} A159841(n,k).
%F A160906 Conjecture: a(2n+1) = A075273(3n).
%F A160906 a(n) = C(3*n+1,n)*Hyper2F1([1,-n],[2*n+2],-1). - _Peter Luschny_, May 19 2015
%F A160906 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
%F A160906 a(n) = [x^n] 1/((1 - 2*x)*(1 - x)^(2*n+1)). - _Ilya Gutkovskiy_, Oct 25 2017
%F A160906 a(n) ~ 3^(3*n + 3/2) / (sqrt(Pi*n) * 2^(2*n + 1)). - _Vaclav Kotesovec_, Oct 25 2017
%F A160906 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(2*n+k,k). - _Seiichi Manyama_, Aug 03 2025
%F A160906 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
%F A160906 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
%F A160906 G.f.: B(x)^2/(1 + (B(x)-1)/3), where B(x) is the g.f. of A005809. - _Seiichi Manyama_, Aug 15 2025
%F A160906 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
%p A160906 A160906 := proc(n) add( A159841(n,k), k=0..n) ; end:
%p A160906 seq(A160906(n), n=0..20) ;
%t A160906 Table[Sum[Binomial[3*n+1, 2*n+k+1], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 25 2017 *)
%o A160906 (Sage)
%o A160906 a = lambda n: binomial(3*n+1,n)*hypergeometric([1,-n],[2*n+2],-1)
%o A160906 [simplify(a(n)) for n in range(21)] # _Peter Luschny_, May 19 2015
%o A160906 (PARI) a(n) = sum(k=0, n, binomial(3*n+1, 2*n+k+1)); \\ _Michel Marcus_, Oct 31 2017
%Y A160906 Cf. A075273, A159841.
%Y A160906 Cf. A005809, A006256, A183160, A226751.
%Y A160906 Cf. A000302, A386811, A386812.
%Y A160906 Cf. A244038, A383326.
%K A160906 nonn,easy
%O A160906 0,2
%A A160906 _R. J. Mathar_, May 29 2009