A107895
Euler transform of n!.
Original entry on oeis.org
1, 1, 3, 9, 36, 168, 961, 6403, 49302, 430190, 4199279, 45326013, 535867338, 6884000262, 95453970483, 1420538043009, 22579098396600, 381704267100888, 6837775526561031, 129377310771795789, 2578101967764973314, 53965231260126083854, 1183813954026245944519
Offset: 0
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EulerTrans := proc(p) local b; b := proc(n) option remember; local d, j;
`if`(n=0,1, add(add(d*p(d),d=numtheory[divisors](j)) *b(n-j),j=1..n)/n) end end:
A107895 := EulerTrans(n->n!): seq(A107895(n),n=0..20);
# After Alois P. Heinz, A000335. [Peter Luschny, Jul 07 2011]
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EulerTrans[p_] := Module[{b}, b[n_] := b[n] = Module[{d, j}, If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]]; b]; A107895 = EulerTrans[Factorial]; Table[A107895[n], {n, 0, 22}] (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *)
A179327
G.f.: Product_{n>=1} 1/(1-x^n)^((n-1)!).
Original entry on oeis.org
1, 1, 2, 4, 11, 37, 167, 925, 6164, 47630, 418227, 4105887, 44529413, 528398441, 6807143686, 94588353184, 1409913624333, 22437692156739, 379673925360239, 6806484898946045, 128862141334488784, 2569079946351669286, 53797816061915662161, 1180533553597621952193
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 37*x^5 + 167*x^6 +...
A(x) = 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)^6*(1-x^5)^24*(1-x^6)^120*...).
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 27*x^4/4 + 121*x^5/5 + 729*x^6/6 + 5041*x^7/7 + 40347*x^8/8 +...+ A062363(n)*x^n/n +...
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)*binomial((i-1)!+j-1, j), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..23); # Alois P. Heinz, Aug 10 2021
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nmax=20; CoefficientList[Series[Product[1/(1-x^k)^((k-1)!),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 14 2015 *)
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{a(n)=polcoeff(exp(sum(m=1,n,sumdiv(m,d,d!)*x^m/m)+x*O(x^n)),n)}
A305867
Expansion of Product_{k>=1} 1/(1 - x^k)^(2*k-1)!!.
Original entry on oeis.org
1, 1, 4, 19, 130, 1120, 11960, 151595, 2230550, 37361755, 701873371, 14610774346, 333746628499, 8298025724194, 223049950124065, 6444634486214748, 199165237980655863, 6555102341516877027, 228905611339161301812, 8452656930719845696590, 329075775511339959533232, 13471099892869946627980017
Offset: 0
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N:= 25:
S:=series(mul((1-x^k)^(-doublefactorial(2*k-1)),k=1..N),x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Jun 12 2018
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nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(2 k - 1)!!, {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (2 d - 1)!!, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 21}]
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