A305868
Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A001147 (double factorial of odd numbers).
Original entry on oeis.org
1, 2, 12, 87, 816, 9194, 122028, 1859460, 32002076, 613890984, 12989299596, 300556859080, 7550646317520, 204687481289946, 5955892982437120, 185158929516065160, 6125200081143892800, 214837724609502834082, 7963817560398871790604, 311101285877489780292000, 12773912991134665452205048
Offset: 1
1/((1 - x) * (1 - x^2)^2 * (1 - x^3)^12 * (1 - x^4)^87 * (1 - x^5)^816 * ... * (1 - x^n)^a(n) * ...) = 1 + 1*x + 1*3*x^2 + 1*3*5*x^3 + 1*3*5*7*x^4 + ... + A001147(k)*x^k + ...
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nn = 21; f[x_] := Product[1/(1 - x^n)^a[n], {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-k x, 1, {k, 1, nn}]), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
nmax = 20; s = ConstantArray[0, nmax]; Do[s[[j]] = j*(2*j - 1)!! - Sum[s[[d]]*(2*j - 2*d - 1)!!, {d, 1, j - 1}], {j, 1, nmax}]; Table[Sum[MoebiusMu[k/d]*s[[d]], {d, Divisors[k]}]/k, {k, 1, nmax}] (* Vaclav Kotesovec, Aug 09 2019 *)
A305869
Expansion of Product_{k>=1} (1 + x^k)^(2*k-1)!!.
Original entry on oeis.org
1, 1, 3, 18, 123, 1098, 11806, 150406, 2218065, 37206485, 699604235, 14572941915, 333037896380, 8283300923765, 222714069807495, 6436292165450693, 198941178161054798, 6548632634238445779, 228705772883364303114, 8446082393596031365629, 328846269698068735291665, 13462627492562640070346824
Offset: 0
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
binomial(doublefactorial(2*i-1), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..23); # Alois P. Heinz, Jun 13 2018
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nmax = 21; CoefficientList[Series[Product[(1 + x^k)^(2 k - 1)!!, {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (2 d - 1)!!, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 21}]
A363254
Product_{n>=1} (1 + a(n)*x^n) = 1 + 1!!*x + 3!!*x^2 + 5!!*x^3 + 7!!*x^4 + ...
Original entry on oeis.org
1, 3, 12, 93, 816, 9264, 122028, 1863849, 32001504, 614224272, 12989299596, 300599511744, 7550646317520, 204694926767040, 5955892801274796, 185160666502244433, 6125200081143892800, 214838236392631067424, 7963817560398871790604, 311101474513327693885056
Offset: 1
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A[m_, n_] := A[m, n] = Which[m == 1, (2 n - 1)!!, m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 20]
A363255
Product_{n>=1} 1 / (1 - a(n)*x^n) = 1 + 1!!*x + 3!!*x^2 + 5!!*x^3 + 7!!*x^4 + ...
Original entry on oeis.org
1, 2, 12, 86, 816, 9126, 122028, 1855802, 32001504, 613558458, 12989299596, 300515004558, 7550646317520, 204680035934550, 5955892801274796, 185157207502788074, 6125200081143892800, 214837212308039658666, 7963817560398871790604, 311101097650387613661510
Offset: 1
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A[m_, n_] := A[m, n] = Which[m == 1, (2 n - 1)!!, m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1] A[m - 1, n - m + 1]]; a[n_] := A[n, n]; a /@ Range[1, 20]
Showing 1-4 of 4 results.
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