A235685
E.g.f.: exp( Sum_{n>=1} H(n) * x^(2*n)/(2*n) ) where H(n) is the n-th harmonic number.
Original entry on oeis.org
1, 1, 12, 370, 21490, 2012346, 276603096, 52410015372, 13086020092860, 4162089324641820, 1642246641208135248, 786999935144858519448, 450157417640763926225496, 302899153448396612425831800, 236824761815538353605549389600, 212892105513043495761147435785040
Offset: 0
E.g.f.: A(x) = 1 + x^2/2! + 12*x^4/4! + 370*x^6/6! + 21490*x^8/8! +...
where
log(A(x)) = x^2/2 + (1+1/2)*x^4/4 + (1+1/2+1/3)*x^6/6 + (1+1/2+1/3+1/4)*x^8/8 + (1+1/2+1/3+1/4+1/5)*x^10/10 + (1+1/2+1/3+1/4+1/5+1/6)*x^12/12 +...
Explicitly,
log(A(x)) = x^2/2! + 9*x^4/4! + 220*x^6/6! + 10500*x^8/8! + 828576*x^10/10! + 97796160*x^12/12! + 16145775360*x^14/14! + 3554072121600*x^16/16! +...
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{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)/(2*k))+x*O(x^n)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(2*n), ", "))
A235776
E.g.f.: exp( Sum_{n>=1} x^(2*n) * Sum_{k=1..n} 1/k^2 ).
Original entry on oeis.org
1, 2, 42, 2000, 170660, 22741992, 4344779208, 1123066676160, 376718037181200, 158895919895100960, 82222168141278271392, 51172838316787466103552, 37687233953299944682503744, 32399590493755848692815785600, 32140659218911596667452247171200
Offset: 0
E.g.f.: A(x) = 1 + 2*x^2/2! + 42*x^4/4! + 2000*x^6/6! + 170660*x^8/8! +...
such that
log(A(x)) = x^2 + (1+1/4)*x^4 + (1+1/4+1/9)*x^6 + (1+1/4+1/9+1/16)*x^8 + (1+1/4+1/9+1/16+1/25)*x^10 + (1+1/4+1/9+1/16+1/25+1/36)*x^12 +...
Explicitly,
log(A(x)) = x^2 + 5/4*x^4 + 49/36*x^6 + 205/144*x^8 + 5269/3600*x^10 + 5369/3600*x^12 + 266681/176400*x^14 +...+ [Sum_{k=1..n} 1/k^2]*x^(2*n) +...
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nmax = 20; CoefficientList[Series[Exp[PolyLog[2,x]/(1-x)], {x, 0, nmax}], x] * (2*Range[0, nmax])! (* Vaclav Kotesovec, Oct 28 2024 *)
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{a(n)=local(A=1); A=exp(sum(m=1, n\2+1, sum(k=1, m, 1/k^2)*x^(2*m))+x*O(x^n)); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(2*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
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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
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