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A156886 a(n) = Sum_{k=0..n} C(n,k)*C(3*n+k,k).

%I A156886 #21 Jun 02 2025 01:23:56
%S A156886 1,5,43,416,4239,44485,475780,5156548,56437231,622361423,6904185523,
%T A156886 76964141600,861408728964,9673849095708,108954068684616,
%U A156886 1230185577016156,13920106205444335,157814104889538739
%N A156886 a(n) = Sum_{k=0..n} C(n,k)*C(3*n+k,k).
%C A156886 a(n)=[x^n](1+5x+9x^2+7x^3+2x^4)^n. The coefficients (1,5,9,7,2) are the 5th row of A029635.
%H A156886 Robert Israel, <a href="/A156886/b156886.txt">Table of n, a(n) for n = 0..937</a>
%H A156886 P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry2/barry231.html">A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.5.4.
%F A156886 From _Peter Bala_, Feb 11 2018: (Start)
%F A156886 a(n) = Sum_{k = 0..n} (-1)^(n-k)*C(n,k)*C(3*n+k,n)*2^k.
%F A156886 a(n) = Sum_{k = 0..n} C(n,k)*C(3*n,k)*2^(n-k),
%F A156886 12*n*(3*n-1)*(3*n-2)*(238*n^2 - 663*n + 457)*a(n) = 2*(150416*n^5 - 644640*n^4 + 1020351*n^3 - 734334*n^2 + 237007*n - 26880)*a(n-1) - (3*n-3)*(3*n-4)*(3*n-5)*(238*n^2 - 187*n + 32)*a(n-2). (End)
%F A156886 a(n) = P_n(0,2*n,3) where P_n(a,b,x) is the n-th Jacobi polynomial with parameters a and b. - _Robert Israel_, Feb 11 2018
%F A156886 a(n) ~ sqrt(1/3 + 11/(12*sqrt(7))) * ((316 + 119*sqrt(7))/54)^n / sqrt(Pi*n). - _Vaclav Kotesovec_, Jan 09 2023
%p A156886 A156886 := proc(n)
%p A156886     add(binomial(n,k)*binomial(3*n+k,k), k = 0..n);
%p A156886 end proc:
%p A156886 seq(A156886(n), n = 0..20); # _Peter Bala_, Feb 11 2018
%t A156886 a[n_] := Sum[ Binomial[n, k] Binomial[3n + k, k], {k, 0, n}]; Array[a, 21, 0] (* _Robert G. Wilson v_, Feb 11 2018 *)
%Y A156886 Cf. A001850, A114496, A029635, A156887.
%K A156886 easy,nonn
%O A156886 0,2
%A A156886 _Paul Barry_, Feb 17 2009