A162650 -id:A162650 - cp's OEIS frontend

A219503 Expansion of e.g.f. Sum_{n>=0} (n+1)^(n-1) * sinh(x)^n / n!.

1, 1, 3, 17, 137, 1457, 19355, 308961, 5766353, 123285153, 2972114803, 79782059249, 2360417058521, 76319622510289, 2677629295171979, 101318751122847297, 4113158120834726049, 178328823993199602241, 8223999403291995520995, 401989145900847087408849

Offset: 0

Paul D. Hanna, Nov 20 2012

nonn

Compare to the LambertW identity: Sum_{n>=0} (n+1)^(n-1)*exp(-n*x)*x^n/n! = exp(x).

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 137*x^4/4! + 1457*x^5/5! +...
where
A(x) = 1 + sinh(x) + 3^1*sinh(x)^2/2! + 4^2*sinh(x)^3/3! + 5^3*sinh(x)^4/4! +...

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A003724, A136630, A162650.

Cf. A238085.

CoefficientList[Series[-LambertW[-Sinh[x]]/Sinh[x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 08 2013 *)

{a(n)=n!*polcoeff(sum(k=0,n,(k+1)^(k-1)*sinh(x + x*O(x^n))^k/k!),n)}
for(n=0,25,print1(a(n),", "))

E.g.f.: LambertW(-sinh(x)) / (-sinh(x)).
a(n) ~ (1+exp(2))^(1/4) * n^(n-1) / (exp(n-1) * log(exp(-1) +sqrt(1+exp(-2)))^(n-1/2)). - Vaclav Kotesovec, Jul 08 2013
a(n) = Sum_{k=0..n} (k+1)^(k-1) * A136630(n,k). - Seiichi Manyama, Feb 15 2025

A381415 E.g.f. A(x) satisfies A(x) = exp( sinh(x * A(x)^2) ).

Original entry on oeis.org

1, 1, 5, 50, 765, 15852, 415441, 13182976, 491502521, 21061603152, 1020066862269, 55107133707232, 3285531022228725, 214295961023511616, 15179005200468020489, 1160334809344169734144, 95214513195493336071537, 8347897781857074205573376, 778804910740650550851809013

Offset: 0

Seiichi Manyama, Feb 23 2025

nonn

Cf. A136630, A162650.

Join[{1}, Table[Sum[(2*n + 1)^(k-1) / (2^k*k!) * Sum[(-1)^(k-j) * (2*j - k)^n * Binomial[k, j], {j, 0, k}], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jul 04 2025 *)

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, (2*n+1)^(k-1)*a136630(n, k));

a(n) = Sum_{k=0..n} (2*n+1)^(k-1) * A136630(n,k).

a(n) ~ s * n^(n-1) / (2*sqrt(1 + r*s^2*sqrt(1 - 4*r^2*s^4)) * exp(n) * r^n), where r = 0.1774317812751606880070098054556619184142424898705... and s = 1.597465072615091018021826608474818660705268320323... are the roots of the system of equations exp(sinh(r*s^2)) = s, 2*r*s^2*cosh(r*s^2) = 1. - Vaclav Kotesovec, Jul 04 2025

A381147 E.g.f. A(x) satisfies A(x) = exp( sinh(x * A(x)) / A(x) ).

Original entry on oeis.org

1, 1, 1, 2, 13, 92, 621, 5112, 56057, 705168, 9480665, 141039648, 2366242693, 43609330624, 864164283269, 18414385180544, 422574196387953, 10374625080684800, 270563138370828465, 7472794772378583552, 218190569313134267517, 6714970997524417977344

Offset: 0

Seiichi Manyama, Feb 15 2025

nonn

Cf. A003724, A162650, A219503.

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, (n-k+1)^(k-1)*a136630(n, k));

a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * A136630(n,k).

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A219503 Expansion of e.g.f. Sum_{n>=0} (n+1)^(n-1) * sinh(x)^n / n!.

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A381415 E.g.f. A(x) satisfies A(x) = exp( sinh(x * A(x)^2) ).

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A381147 E.g.f. A(x) satisfies A(x) = exp( sinh(x * A(x)) / A(x) ).

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