A396238 - OEIS

%I #20 May 21 2026 06:46:00

%S 1,4,22,130,789,4848,29975,185953,1155696,7190013,44757663,278707048,

%T 1735840329,10812295670,67352305826,419567747884,2613727621445,

%U 16282589303341,101435368987532,631912517383722,3936637423973502,24524174461033569,152779004164967032

%N a(n) = Sum_{i=0..n} Sum_{j=0..i} Sum_{k=0..j} binomial(n+i,2*i) * binomial(i+j,2*j) * binomial(j+k,2*k).

%H Seiichi Manyama, <a href="/A396238/b396238.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (15,-83,220,-303,220,-83,15,-1).

%F G.f.: (1-x) * ((1-x)^2 - x) * (((1-x)^2 - x)^2 - x*(1-x)^2) / ((((1-x)^2 - x)^2 - x*(1-x)^2)^2 - x*(1-x)^2*((1-x)^2 - x)^2).

%F a(n) = 15*a(n-1) - 83*a(n-2) + 220*a(n-3) - 303*a(n-4) + 220*a(n-5) - 83*a(n-6) + 15*a(n-7) - a(n-8).

%F G.f.: 1 / ( (1-x) * (1-B(x)) * (1-B(B(x))) * (1-B(B(B(x)))) ), where B(x) = x/(1-x)^2.

%F G.f.: (C(x)/x)^(1/2), where C(x) is the g.f. of A396094.

%o (PARI) a(n) = sum(i=0, n, sum(j=0, i, sum(k=0, j, binomial(n+i, 2*i)*binomial(i+j, 2*j)*binomial(j+k, 2*k))));

%Y Cf. A122367, A166482.

%Y Cf. A396094.

%K nonn

%O 0,2

%A _Seiichi Manyama_, May 19 2026