A308060
G.f. A(x) satisfies: A(x) = x * exp(Sum_{i>=1} Sum_{j>=1} (-1)^(j+1)*A(x^(i*j))/j).
Original entry on oeis.org
1, 1, 2, 5, 11, 26, 65, 161, 412, 1074, 2841, 7599, 20582, 56202, 154760, 429052, 1196802, 3356107, 9456737, 26760173, 76017365, 216693521, 619663800, 1777141141, 5110235884, 14730604451, 42557910762, 123210505445, 357403386959, 1038616488923, 3023329186466, 8814593734152
Offset: 1
-
terms = 32; A[] = 0; Do[A[x] = x Exp[Sum[Sum[(-1)^(j + 1) A[x^(i j)]/j, {j, 1, terms}], {i, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
a[n_] := a[n] = SeriesCoefficient[x Product[Product[(1 + x^(i j))^a[j], {j, 1, n - 1}], {i, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 32}]
A308153
G.f.: x * Product_{j>=1, k>=1} 1/(1 - a(j)*x^(j*k)).
Original entry on oeis.org
1, 1, 3, 7, 19, 47, 134, 357, 1031, 2912, 8612, 25007, 75378, 223884, 683915, 2067578, 6376800, 19503546, 60749341, 187592661, 587938043, 1831377952, 5773159368, 18092820941, 57328904204, 180657986051, 574735018826, 1820143698295, 5810522774503, 18473074695503
Offset: 1
-
A:= proc(n) option remember; series(x*`if`(n=1, 1, mul(mul(
1/(1-a(j)*x^(j*k)), k=1..(n-1)/j), j=1..n-1)), x, n+1)
end:
a:= n-> coeff(A(n), x, n):
seq(a(n), n=1..35); # Alois P. Heinz, May 14 2019
-
a[n_] := a[n] = SeriesCoefficient[x Product[Product[1/(1 - a[j] x^(j k)), {k, 1, n - 1}], {j, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 30}]
A308152
G.f.: x * Product_{j>=1, k>=1} ((1 + x^(j*k))/(1 - x^(j*k)))^a(j).
Original entry on oeis.org
1, 2, 8, 32, 138, 612, 2864, 13712, 67416, 337482, 1716208, 8837392, 45997032, 241571408, 1278625480, 6813568656, 36524390042, 196820310100, 1065583770168, 5793299764208, 31615962617272, 173131117881312, 951040865156928, 5239171609158304, 28937688613453048
Offset: 1
-
a[n_] := a[n] = SeriesCoefficient[x Product[Product[((1 + x^(j k))/(1 - x^(j k)))^a[j], {k, 1, n - 1}], {j, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 25}]
terms = 25; A[] = 0; Do[A[x] = x Exp[2 Sum[Sum[A[x^(i (2 j - 1))]/(2 j - 1), {j, 1, terms}], {i, 1,terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
Showing 1-3 of 3 results.