A036249 -id:A036249 - cp's OEIS frontend

A363578 G.f. satisfies A(x) = exp( Sum_{k>=1} ((-2)^k + A(x^k)) * x^k/k ).

1, -1, 2, -2, 4, -6, 13, -20, 38, -65, 129, -228, 435, -794, 1528, -2833, 5421, -10189, 19561, -37091, 71247, -135973, 261879, -502303, 969181, -1866210, 3608664, -6970576, 13504298, -26152744, 50758711, -98515611, 191517618, -372404560, 725061378

Offset: 0

Seiichi Manyama, Jun 10 2023

sign

Cf. A001678, A036249, A362389, A363541, A363579.

Cf. A363580.

seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, ((-2)^k+subst(A, x, x^k))*x^k/k)+x*O(x^n))); Vec(A);

A(x) = B(x)/(1 + 2*x) where B(x) is the g.f. of A363580.
A(x) = Sum_{k>=0} a(k) * x^k = 1/(1+2*x) * 1/Product_{k>=0} (1-x^(k+1))^a(k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( (-2)^k + Sum_{d|k} d * a(d-1) ) * a(n-k).

A363579 G.f. satisfies A(x) = exp( Sum_{k>=1} ((-3)^k + A(x^k)) * x^k/k ).

Original entry on oeis.org

1, -2, 5, -11, 27, -70, 188, -502, 1355, -3712, 10269, -28546, 79777, -224153, 632581, -1791644, 5091109, -14510079, 41464784, -118773034, 340950420, -980660721, 2825700987, -8155455450, 23573749136, -68236663474, 197774787066, -573915774310, 1667300177595

Offset: 0

Seiichi Manyama, Jun 10 2023

sign

Cf. A001678, A036249, A362389, A363541, A363578.

Cf. A363581.

seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, ((-3)^k+subst(A, x, x^k))*x^k/k)+x*O(x^n))); Vec(A);

A(x) = B(x)/(1 + 3*x) where B(x) is the g.f. of A363581.
A(x) = Sum_{k>=0} a(k) * x^k = 1/(1+3*x) * 1/Product_{k>=0} (1-x^(k+1))^a(k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( (-3)^k + Sum_{d|k} d * a(d-1) ) * a(n-k).

A308227 G.f.: (x/(1 - x)) * Product_{k>=1} ((1 + x^k)/(1 - x^k))^a(k).

Original entry on oeis.org

1, 3, 11, 47, 217, 1065, 5453, 28789, 155633, 857207, 4793103, 27136555, 155249971, 896133487, 5212477023, 30522169103, 179777122393, 1064411910393, 6331361864657, 37817265028841, 226731778956181, 1363993567341257, 8231111557650837, 49812263080757845

Offset: 1

Ilya Gutkovskiy, May 16 2019

nonn

Cf. A029856, A036249, A052829, A073075.

a[n_] := a[n] = SeriesCoefficient[x/(1 - x) Product[((1 + x^k)/(1 - x^k))^a[k], {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 24}]
terms = 24; A[] = 0; Do[A[x] = x Exp[Sum[2 A[x^(2 k - 1)]/(2 k - 1) + x^k/k, {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

G.f. A(x) satisfies: A(x) = x * exp(Sum_{k>=1} 2*A(x^(2*k-1))/(2*k - 1) + x^k/k).

cp's OEIS Frontend

A363578 G.f. satisfies A(x) = exp( Sum_{k>=1} ((-2)^k + A(x^k)) * x^k/k ).

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A363579 G.f. satisfies A(x) = exp( Sum_{k>=1} ((-3)^k + A(x^k)) * x^k/k ).

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Author

Keywords

Crossrefs

Programs

PARI

Formula

A308227 G.f.: (x/(1 - x)) * Product_{k>=1} ((1 + x^k)/(1 - x^k))^a(k).

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Author

Keywords

Crossrefs

Programs

Mathematica

Formula