A373740 -id:A373740 - cp's OEIS frontend

A376512 Expansion of e.g.f. exp(x^2 * (1 + x)).

1, 0, 2, 6, 12, 120, 480, 2520, 21840, 120960, 937440, 8316000, 60540480, 570810240, 5465940480, 49037788800, 523588665600, 5504686387200, 57816850291200, 678823104960000, 7844848544332800, 93064133530368000, 1184800751111577600, 14967781957781452800

Offset: 0

Seiichi Manyama, Sep 25 2024

Cf. A047974, A376513.

Cf. A118589, A227937, A373740.

a(n) = n!*sum(k=0, n\2, binomial(k, n-2*k)/k!);

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(k,n-2*k)/k!.
a(n) = (n-1) * (2*a(n-2) + 3*(n-2)*a(n-3)).
a(n) ~ 3^(n/3 - 1/2) * exp(4/81 - 2*3^(-7/3)*n^(1/3) + 3^(-2/3)*n^(2/3) - 2*n/3) * n^(2*n/3) * (1 + 223/(3^(20/3)*n^(1/3))). - Vaclav Kotesovec, Sep 26 2024

A373741 Expansion of e.g.f. exp(x^2/2 * (1 + x)^3).

Original entry on oeis.org

1, 0, 1, 9, 39, 150, 1365, 13545, 105945, 918540, 10603845, 127806525, 1468823895, 18253765530, 257397445305, 3770163121725, 55637459903025, 866703333295800, 14468243658093225, 250223925107581425, 4426399346291497575, 81488489549760042750

Offset: 0

Seiichi Manyama, Jun 16 2024

nonn

Cf. A361567, A373740.

Cf. A116090.

With[{nn=30},CoefficientList[Series[Exp[x^2/2 (1+x)^3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 26 2025 *)

a(n) = n!*sum(k=0, n\2, binomial(3*k, n-2*k)/(2^k*k!));

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(3*k,n-2*k)/(2^k * k!).

a(n) = (n-1)/2 * (2*a(n-2) + 9*(n-2)*a(n-3) + 12*(n-2)*(n-3)*a(n-4) + 5*(n-2)*(n-3)*(n-4)*a(n-5)).

cp's OEIS Frontend

A376512 Expansion of e.g.f. exp(x^2 * (1 + x)).

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