A235685 -id:A235685 - cp's OEIS frontend

A235385 E.g.f.: exp( Sum_{n>=1} H(n) * x^(2n-1)/(2n-1) ) where H(n) is the n-th harmonic number.

1, 1, 1, 4, 13, 75, 415, 3160, 24545, 233509, 2323165, 26599780, 321545365, 4312503655, 61219938915, 942271981240, 15340303899265, 266671144108265, 4892612440317145, 94840103781865060, 1934826541931748925, 41387703314570495875, 928953515444722956775, 21738929496091877729400

Offset: 0

Paul D. Hanna, Jan 08 2014

nonn

Compare to: exp( Sum_{n>=1} x^(2*n-1)/(2*n-1) ) = sqrt(1-x^2)/(1-x).

			E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 75*x^5/5! +...
where
log(A(x)) = x + (1+1/2)*x^3/3 + (1+1/2+1/3)*x^5/5 + (1+1/2+1/3+1/4)*x^7/7 + (1+1/2+1/3+1/4+1/5)*x^9/9 + (1+1/2+1/3+1/4+1/5+1/6)*x^11/11 +...
Explicitly,
log(A(x)) = x + 1/2*x^3 + 11/30*x^5 + 25/84*x^7 + 137/540*x^9 + 49/220*x^11 + 363/1820*x^13 + 761/4200*x^15 +...
Equivalently,
log(A(x)) = x + 3*x^3/3! + 44*x^5/5! + 1500*x^7/7! + 92064*x^9/9! + 8890560*x^11/11! + 1241982720*x^13/13! + 236938141440*x^15/15! +...

Cf. A235685, A087761.

{H(n)=sum(k=1,n,1/k)}
{a(n)=local(A=1);A=exp(sum(k=1,n\2+1,H(k)*x^(2*k-1)/(2*k-1))+x*O(x^n));n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

A336289 a(0) = 1; a(n) = n! * Sum_{k=1..n} binomial(n-1,k-1) * (k-1)! * H(k) * a(n-k) / (n-k)!, where H(k) is the k-th harmonic number.

Original entry on oeis.org

1, 1, 5, 55, 1054, 31046, 1299386, 73211510, 5338080280, 488727800664, 54865512897432, 7408400404206792, 1184230737883333680, 221121985937352261360, 47683177920267470877648, 11758982455716373002624816, 3287966057434181416523799936

Offset: 0

Ilya Gutkovskiy, Jul 16 2020

nonn

Cf. A001008, A002805, A074707, A087761, A235385, A235685, A336290.

a[0] = 1; a[n_] := a[n] = n! Sum[Binomial[n - 1, k - 1] (k - 1)! HarmonicNumber[k] a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[Exp[Sum[HarmonicNumber[k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2
nmax = 16; CoefficientList[Series[Exp[Log[1 - x]^2/2 + PolyLog[2, x]], {x, 0, nmax}], x] Range[0, nmax]!^2

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} H(n) * x^n / n).

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(log(1 - x)^2 / 2 + polylog(2,x)).

cp's OEIS Frontend

A235385 E.g.f.: exp( Sum_{n>=1} H(n) * x^(2n-1)/(2n-1) ) where H(n) is the n-th harmonic number.

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A336289 a(0) = 1; a(n) = n! * Sum_{k=1..n} binomial(n-1,k-1) * (k-1)! * H(k) * a(n-k) / (n-k)!, where H(k) is the k-th harmonic number.

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cp's OEIS Frontend

A235385 E.g.f.: exp( Sum_{n>=1} H(n) * x^(2*n-1)/(2*n-1) ) where H(n) is the n-th harmonic number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A336289 a(0) = 1; a(n) = n! * Sum_{k=1..n} binomial(n-1,k-1) * (k-1)! * H(k) * a(n-k) / (n-k)!, where H(k) is the k-th harmonic number.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A235385 E.g.f.: exp( Sum_{n>=1} H(n) * x^(2n-1)/(2n-1) ) where H(n) is the n-th harmonic number.