A373680 -id:A373680 - cp's OEIS frontend

A373685 Expansion of e.g.f. exp(x / (1 - x^3)^2) / (1 - x^3).

1, 1, 1, 7, 73, 301, 1561, 32131, 306097, 2062873, 43102801, 720515071, 7245589561, 136364378437, 3259345980073, 47903339552251, 873735377165281, 25383884535029041, 515592396859327777, 10003196649764818423, 316630570292623967401, 8359224513085985870941

Offset: 0

Seiichi Manyama, Jun 13 2024

nonn

Cf. A373680, A373684.

a(n) = n!*sum(k=0, n\3, binomial(2*n-5*k, k)/(n-3*k)!);

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(2*n-5*k,k)/(n-3*k)!.

a(n) == 1 (mod 6).

A373707 Expansion of e.g.f. exp(x * (1 + x^3)^2).

Original entry on oeis.org

1, 1, 1, 1, 49, 241, 721, 6721, 124321, 913249, 4243681, 94818241, 1640604241, 14642181841, 131026944049, 3669304504321, 62536989802561, 627395160826561, 10818406189690561, 308036857749752449, 5219006583104930161, 65146235714284117681

Offset: 0

Seiichi Manyama, Jun 14 2024

nonn

Cf. A361278, A373578, A373708.
Cf. A190875, A373522, A373523.
Cf. A373680.

a(n) = n!*sum(k=0, 2*n\7, binomial(2*n-6*k, k)/(n-3*k)!);

a(n) = n! * Sum_{k=0..floor(2*n/7)} binomial(2*n-6*k,k)/(n-3*k)!.
a(n) == 1 (mod 48).
a(n) = a(n-1) + 8*(n-1)*(n-2)*(n-3)*a(n-4) + 7*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*a(n-7).

cp's OEIS Frontend

A373685 Expansion of e.g.f. exp(x / (1 - x^3)^2) / (1 - x^3).

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