A381284 -id:A381284 - cp's OEIS frontend

A381346 Expansion of e.g.f. 1/( 1 - sinh(sqrt(2)*x) / sqrt(2) ).

1, 1, 2, 8, 40, 244, 1808, 15632, 154240, 1712656, 21132032, 286800128, 4246266880, 68108302144, 1176458774528, 21772909267712, 429818456473600, 9015349812633856, 200218257664704512, 4693597812326094848, 115820240623410872320, 3000905720793597113344

Offset: 0

Seiichi Manyama, Feb 20 2025

nonn

Cf. A191277, A381284, A381285, A381286, A381347.

Cf. A136630.

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*2^((n-k)/2)*a136630(n, k));

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 2^k * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * 2^((n-k)/2) * A136630(n,k).
a(n) ~ sqrt(Pi/3) * 2^(n/2 + 1) * n^(n + 1/2) / (arcsinh(sqrt(2))^(n+1) * exp(n)). - Vaclav Kotesovec, Apr 19 2025

A381347 Expansion of e.g.f. 1/( 1 - sin(sqrt(2)*x) / sqrt(2) ).

Original entry on oeis.org

1, 1, 2, 4, 8, 4, -112, -1184, -9088, -59504, -310528, -643136, 14701568, 323581504, 4554426368, 51666451456, 458243735552, 2004840714496, -37024075153408, -1386061762251776, -29290212127670272, -483475390212586496, -6224109737622372352, -45231727252157947904

Offset: 0

Seiichi Manyama, Feb 20 2025

sign

Cf. A191277, A381284, A381285, A381286, A381346.

Cf. A136630, A263249.

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*(-2)^((n-k)/2)*a136630(n, k));

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-2)^k * binomial(n,2*k+1) * a(n-2*k-1).

a(n) = Sum_{k=0..n} k! * (-2)^((n-k)/2) * A136630(n,k)

cp's OEIS Frontend

A381346 Expansion of e.g.f. 1/( 1 - sinh(sqrt(2)*x) / sqrt(2) ).

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