A305870
Product_{n>=1} (1 + x^n)^a(n) = g.f. of A001147 (double factorial of odd numbers).
Original entry on oeis.org
1, 3, 12, 90, 816, 9206, 122028, 1859550, 32002076, 613891800, 12989299596, 300556868286, 7550646317520, 204687481411974, 5955892982437120, 185158929517924710, 6125200081143892800, 214837724609534836158, 7963817560398871790604, 311101285877490394183800, 12773912991134665452205048
Offset: 1
(1 + x) * (1 + x^2)^3 * (1 + x^3)^12 * (1 + x^4)^90 * (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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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= proc(n) option remember; doublefactorial(2*n-1)-b(n, n-1) end:
seq(a(n), n=1..23); # Alois P. Heinz, Jun 13 2018
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nn = 21; f[x_] := Product[(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
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}]
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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