A296162
a(n) = [x^n] Product_{k>=1} ((1 - x^(3*k))/(1 - x^k))^n.
Original entry on oeis.org
1, 1, 5, 19, 89, 406, 1913, 9073, 43505, 209971, 1019390, 4971781, 24343037, 119579006, 589050663, 2908727839, 14393759457, 71360342129, 354372852011, 1762416036422, 8776797353574, 43761058841638, 218431753457637, 1091385769314272, 5458068218285909, 27319061357620056
Offset: 0
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Table[SeriesCoefficient[Product[((1 - x^(3 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[(1 + x^k + x^(2 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
(* Calculation of constant d: *) With[{k = 3}, 1/r /. FindRoot[{s == QPochhammer[(r*s)^k] / QPochhammer[r*s], k*(-(s*QPochhammer[r*s]*(Log[1 - (r*s)^k] + QPolyGamma[0, 1, (r*s)^k]) / Log[(r*s)^k]) + (r*s)^k * Derivative[0, 1][QPochhammer][(r*s)^k, (r*s)^k]) == s*QPochhammer[r*s] + s^2*(-(QPochhammer[r*s]*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)
A304459
Coefficient of x^n in Product_{k>=1} (1+x^k)^(n^3).
Original entry on oeis.org
1, 1, 36, 3681, 770576, 276218900, 151479085752, 117975860569973, 123825991870849088, 168480096257782525419, 288418999876101261408100, 606652152400218992684850772, 1537897976017806908644807294656, 4624364862288125600795358272563097
Offset: 0
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nmax = 20; Table[SeriesCoefficient[Product[(1+x^k)^(n^3), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 20; Table[SeriesCoefficient[(QPochhammer[-1, x]/2)^(n^3), {x, 0, n}], {n, 0, nmax}]
A386729
a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} (1 + x^k)^(k^3) is the g.f. of A248882.
Original entry on oeis.org
1, 1, 17, 154, 1377, 13276, 127862, 1249746, 12321121, 122287798, 1220492192, 12235940113, 123133325382, 1243080020352, 12583773308102, 127688996851804, 1298370095026017, 13226355435367992, 134955405683954234, 1379032238329708409, 14110075394718902752, 144544237021110644340
Offset: 0
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Table[SeriesCoefficient[Product[(1+x^k)^(n*k^3), {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Exp[n*Sum[Sum[(-1)^(k/d + 1)*d^4, {d, Divisors[k]}]*x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 25}]
A341395
Coefficient of x^(2*n) in (-1 + Product_{k>=1} (1 + x^k)^k)^n.
Original entry on oeis.org
1, 2, 14, 92, 662, 4872, 36578, 278161, 2135902, 16522967, 128574734, 1005321616, 7891885382, 62160038813, 491003317483, 3888045701232, 30854283708670, 245315312649653, 1953735732991919, 15583347966328833, 124463844976490422, 995305632560023009, 7968042676400949882
Offset: 0
Cf.
A026007,
A257675,
A270913,
A270922,
A324595,
A341384,
A341385,
A341386,
A341387,
A341388,
A341390,
A341391,
A341393,
A341394.
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g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
`if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
end:
b:= proc(n, k) option remember; `if`(k<2, g(n+1), (q->
add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..22); # Alois P. Heinz, Feb 10 2021
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Join[{1}, Table[SeriesCoefficient[(-1 + Product[(1 + x^k)^k, {k, 1, 2 n}])^n, {x, 0, 2 n}], {n, 1, 22}]]
A[n_, k_] := A[n, k] = If[n == 0, 1, k Sum[A[n - j, k] Sum[(-1)^(j/d + 1) d^2, {d, Divisors[j]}], {j, 1, n}]/n]; T[n_, k_] := Sum[(-1)^i Binomial[k, i] A[n, k - i], {i, 0, k}]; Table[T[2 n, n], {n, 0, 22}]
A369725
Maximal coefficient of ( (1 + x) * (1 + x^2) * (1 + x^3) * ... * (1 + x^n) )^n.
Original entry on oeis.org
1, 1, 4, 62, 4658, 1585430, 2319512420, 14225426190522, 361926393013029354, 37883831957216781279561, 16231015449888734994721650504, 28330316118212024049511095643949434, 200866780133770636272812495083578779133456, 5771133366532656054669819186294860881172794669798
Offset: 0
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a:= n-> max(coeffs(expand(mul(1+x^k, k=1..n)^n))):
seq(a(n), n=0..14); # Alois P. Heinz, Jan 30 2024
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Table[Max[CoefficientList[Product[(1 + x^k)^n, {k, 1, n}], x]], {n, 0, 13}]
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a(n) = vecmax(Vec(prod(k=1, n, (1+x^k))^n)); \\ Michel Marcus, Jan 30 2024
A382950
a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k + y^k)^n.
Original entry on oeis.org
1, 0, 6, 102, 1342, 20030, 306852, 4783534, 75873934, 1220259306, 19837742836, 325375411438, 5376744428812, 89412908941096, 1494992390431000, 25114561595879252, 423649216254936110, 7172523302899053230, 121828099966104173892, 2075321708914763792740
Offset: 0
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Table[SeriesCoefficient[Product[(1 + x^k + y^k)^n, {k, 1, n}], {x, 0, n}, {y, 0, n}], {n, 0, 19}]
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