A396286 - OEIS

A396286

a(n) = Sum_{k=0..n} binomial(n+3*k,4*k) * (binomial(n+4*k,k) - binomial(n+4*k,k-1)).

1, 5, 61, 837, 12101, 180405, 2743189, 42289877, 658586965, 10336163925, 163218825045, 2590244957525, 41275966942549, 660016673641813, 10585031873029461, 170189634655925589, 2742431029234324821, 44277565440708269397, 716111715656137069909, 11599681231606159267157

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OFFSET

0,2

LINKS

Table of n, a(n) for n=0..19.

FORMULA

a(n) = A397386(n) - A397385(n).

For m >= 0 and any constants r, s, define a_{m,r,s}(n) = Sum_{k=0..n} binomial(n+(m-1)*k,m*k) * (r*binomial(n+m*k,k) - s*binomial(n+m*k,k-1)). A_{m,r,s}(x) = Sum_{n>=0} a_{m,r,s}(n)*x^n = t*(1+t^m)*(r+(r+s)*t^(m-1)-s*t^m)/((1+t^(m-1))*(1+(m+1)*t^m-m*t^(m+1))), where t = t(x) satisfies t = 1 + x*t*(1+t^m).

PROG

(PARI) a(n) = sum(k=0, n, binomial(n+3*k, 4*k)*(binomial(n+4*k, k)-binomial(n+4*k, k-1)));

CROSSREFS

Cf. A116363, A394186, A395801.

Cf. A349311, A397385, A397386.

Sequence in context: A189444 A373106 A217811 * A217280 A236955 A064328

Adjacent sequences: A396283 A396284 A396285 * A396287 A396288 A396289

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Jun 23 2026

STATUS

approved