A381386 -id:A381386 - cp's OEIS frontend

A381387 E.g.f. A(x) satisfies A(x) = 1/( 1 - sinh(x * A(x)) )^2.

1, 2, 14, 182, 3520, 91002, 2954400, 115638014, 5303063552, 278979672050, 16565016146176, 1095997724407302, 79966475806040064, 6379010456725968362, 552344502268240535552, 51595059327775839277646, 5171865567269556457308160, 553764742712510134123863522

Offset: 0

Seiichi Manyama, Feb 22 2025

nonn

Cf. A136630, A381386.

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

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381386.
a(n) = 2 * Sum_{k=0..n} k! * binomial(2*n+k+2,k)/(2*n+k+2) * A136630(n,k).
E.g.f.: (1/x) * Series_Reversion( x*(1 - sinh(x))^2 ).

A381477 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * A(x)^2 * cosh(x * A(x)^2) ).

Original entry on oeis.org

1, 1, 6, 75, 1440, 37445, 1231440, 49037527, 2294425728, 123393443049, 7500623201280, 508577491719011, 38057966976387072, 3115680296111519149, 277005128553759191040, 26579020362900758232495, 2737628961211699538657280, 301278578823933606439917137, 35281158151116225085977526272

Offset: 0

Seiichi Manyama, Feb 24 2025

nonn

As stated in the comment of A185951, A185951(n,0) = 0^n.

Cf. A185951, A364985, A381386.

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n+k+1, k)/(2*n+k+1)*a185951(n, k));

a(n) = Sum_{k=0..n} k! * binomial(2*n+k+1,k)/(2*n+k+1) * A185951(n,k).

E.g.f.: ( (1/x) * Series_Reversion( x*(1 - x*cosh(x))^2 ) )^(1/2).

cp's OEIS Frontend

A381387 E.g.f. A(x) satisfies A(x) = 1/( 1 - sinh(x * A(x)) )^2.

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