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A331323 a(n) = [x^n] (1 - 2*x)/(1 - 8*x + 4*x^2)^(3/2).

%I A331323 #36 Mar 23 2022 07:37:30
%S A331323 1,10,90,772,6430,52524,423220,3375880,26720118,210195100,1645295212,
%T A331323 12825551160,99633196780,771702434104,5961969066600,45958506432016,
%U A331323 353585912577190,2715647948258940,20824876515839932,159474192002499160,1219708190630800836,9318143974952519080
%N A331323 a(n) = [x^n] (1 - 2*x)/(1 - 8*x + 4*x^2)^(3/2).
%H A331323 Seiichi Manyama, <a href="/A331323/b331323.txt">Table of n, a(n) for n = 0..1000</a>
%F A331323 a(n) = (-2)^n*Sum_{k=0..n} A331431(n, k)/(-2)^k.
%F A331323 a(n) = (10*n*a(n-1) + 20*(1-n)*a(n-2) + 8*(n-1)*a(n-3))/n.
%F A331323 a(n) = 2^(n-1)*(n+1)*(n*hypergeom([1-n, n+2], [2], -1/2) + 2*hypergeom([-n, n+1], [1], -1/2)).
%F A331323 a(n) = Sum_{k=0..n} 3^k * (k+1) * binomial(n+1,k+1)^2. - _Seiichi Manyama_, Jan 20 2020
%F A331323 a(n) = (n + 1)^2*hypergeom([-n, -n], [2], 3). - _Peter Luschny_, Jan 20 2020
%F A331323 n * (2*n-1) * a(n) = 2 * (8 * n^2 - 3) * a(n-1) - 4 * n * (2*n+1) * a(n-2) for n>1. - _Seiichi Manyama_, Jan 25 2020
%p A331323 gf := (1-2*x)/(4*x^2-8*x+1)^(3/2): ser := series(gf, x, 32):
%p A331323 seq(coeff(ser, x, n), n=0..21); # Or:
%p A331323 a := proc(n) option remember; if n<3 then [1, 10, 90][n+1] else
%p A331323 (10*n*a(n-1) + 20*(1-n)*a(n-2) + 8*(n-1)*a(n-3))/n fi end:
%p A331323 seq(a(n), n=0..21);
%t A331323 a[n_] := Sum[3^k * (k + 1) * Binomial[n + 1, k + 1]^2, {k, 0, n}]; Array[a, 22, 0] (* _Amiram Eldar_, Jan 20 2020 *)
%o A331323 (PARI) {a(n) = 2^n*sum(k=0, n, (n+k+1)*binomial(n, k)*binomial(n+k, k)/2^k)} \\ _Seiichi Manyama_, Jan 18 2020
%o A331323 (PARI) N=20; x='x+O('x^N); Vec((1-2*x)/(4*x^2-8*x+1)^(3/2)) \\ _Seiichi Manyama_, Jan 18 2020
%o A331323 (PARI) {a(n) = sum(k=0, n, 3^k*(k+1)*binomial(n+1, k+1)^2)} \\ _Seiichi Manyama_, Jan 20 2020
%o A331323 (Magma) [(&+[3^k*(k+1)*Binomial(n+1,k+1)^2: k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Mar 22 2022
%o A331323 (Sage) [sum(2^(n-k)*(n+k+1)*binomial(2*k,k)*binomial(n+k,2*k)  for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Mar 22 2022
%Y A331323 Column 5 of A331511.
%Y A331323 Cf. A331431.
%K A331323 nonn
%O A331323 0,2
%A A331323 _Peter Luschny_, Jan 18 2020