A326865
G.f.: Product_{k>=1} (1 + x^k/k^3) = Sum_{n>=0} a(n)*x^n/n!^3.
Original entry on oeis.org
1, 1, 1, 35, 728, 48824, 7170984, 1418111064, 479963197440, 235727037775872, 170423013422592000, 163854260184125952000, 214343327259234349056000, 360795240553638133592064000, 778954481701636984110452736000, 2095759092922096320907078496256000
Offset: 0
-
b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..20); # Alois P. Heinz, Jul 27 2023
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nmax = 20; CoefficientList[Series[Product[(1+x^k/k^3), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!^3
A336306
a(n) = (n!)^n * [x^n] Product_{n>=1} (1 + x^k/k^n).
Original entry on oeis.org
1, 1, 1, 35, 5392, 35462624, 15419509448256, 445352317449860352384, 1733058447330128629281872412672, 1124641798042952855847954946807366969982976, 155064212713307814902013200520441969883490549760000000000
Offset: 0
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b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 b(n$3):
seq(a(n), n=0..12); # Alois P. Heinz, Jul 27 2023
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Table[(n!)^n SeriesCoefficient[Product[(1 + x^k/k^n), {k, 1, n}], {x, 0, n}], {n, 0, 10}]
A346312
Sum_{n>=0} a(n) * x^n / (n!)^2 = Product_{n>=1} (1 - x^n / n^2).
Original entry on oeis.org
1, -1, -1, 5, 28, 724, 36, 220716, -1255680, 110979072, 2530310400, 1193835283200, -24457819622400, 21656019855744000, 899271273253248000, 474367063601421849600, 45822442913828595302400, 28365278076547150440038400, 2614371018285307258994688000
Offset: 0
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nmax = 18; CoefficientList[Series[Product[(1 - x^k/k^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[(Binomial[n, k] k!)^2 Sum[1/(k/d)^(2 d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
A346313
Sum_{n>=0} a(n) * x^n / (n!)^2 = Product_{n>=1} 1 / (1 + (-x)^n / n^2).
Original entry on oeis.org
1, 1, 3, 31, 496, 12576, 444736, 22056448, 1406058816, 114618828096, 11405077216704, 1385889578069184, 198961869847145472, 33725910553646229504, 6594186368339077238784, 1487133154121568112705536, 379990326228614750079369216, 110013397755650063836228435968
Offset: 0
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nmax = 17; CoefficientList[Series[Product[1/(1 + (-x)^k/k^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[(-1)^k (Binomial[n, k] k!)^2 Sum[(-1)^d/(k/d)^(2 d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
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