This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385563 #40 Aug 25 2025 04:38:48
%S A385563 1,9,60,360,2055,11403,62132,334260,1781415,9425295,49581576,
%T A385563 259601004,1353939405,7038232425,36484340400,188665670880,
%U A385563 973545780195,5014258620075,25783103206100,132378800689800,678768332410245,3476164133573505,17782899991147500
%N A385563 Expansion of 1/((1-x) * (1-5*x))^(3/2).
%H A385563 Paolo Xausa, <a href="/A385563/b385563.txt">Table of n, a(n) for n = 0..1000</a>
%F A385563 n*a(n) = (6*n+3)*a(n-1) - 5*(n+1)*a(n-2) for n > 1.
%F A385563 a(n) = (1/4)^n * Sum_{k=0..n} 5^k * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
%F A385563 a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
%F A385563 a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
%F A385563 a(n) = binomial(n+2,2) * A002212(n+1).
%F A385563 a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
%F A385563 a(n) = Sum_{k=0..n} (3/2)^k * (-5/6)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).
%F A385563 a(n) ~ sqrt(n) * 5^(n + 3/2) / (4*sqrt(Pi)). - _Vaclav Kotesovec_, Aug 21 2025
%t A385563 Module[{a, n}, RecurrenceTable[{a[n] == ((6*n+3)*a[n-1] - 5*(n+1)*a[n-2])/n, a[0] == 1, a[1] == 9}, a, {n, 0, 25}]] (* _Paolo Xausa_, Aug 21 2025 *)
%o A385563 (PARI) my(N=30, x='x+O('x^N)); Vec(1/((1-x)*(1-5*x))^(3/2))
%Y A385563 Partial sums of A383254.
%Y A385563 Cf. A331516, A385716.
%Y A385563 Cf. A385728, A385813.
%Y A385563 Cf. A002212, A383949.
%K A385563 nonn,changed
%O A385563 0,2
%A A385563 _Seiichi Manyama_, Aug 19 2025