A363546 -id:A363546 - cp's OEIS frontend

A363541 G.f. satisfies A(x) = exp( Sum_{k>=1} (3^k + A(x^k)) * x^k/k ).

1, 4, 17, 73, 324, 1469, 6838, 32490, 157398, 775010, 3870690, 19567202, 99957231, 515250057, 2676884745, 14002926871, 73693381322, 389904743248, 2072794614996, 11066421965311, 59310040841395, 318978744562253, 1720962766007827

Offset: 0

Seiichi Manyama, Jun 09 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A001678, A036249, A362389.

Cf. A363507, A363543, A363546.

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 A363546.
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).

A363545 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^k) * x^k/(k * (1 - 2*x^k)) ).

Original entry on oeis.org

1, 1, 4, 14, 54, 206, 823, 3312, 13619, 56643, 238569, 1014443, 4352038, 18809992, 81843021, 358186642, 1575810191, 6965004499, 30914431131, 137736012285, 615785575785, 2761693248028, 12421390811559, 56016050571825, 253228531426237

Offset: 0

Seiichi Manyama, Jun 09 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A052855, A198518, A363546, A363580, A363581.

Cf. A362389.

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

A(x) = (1 - 2*x) * B(x) where B(x) is the g.f. of A362389.

a(n) = A362389(n) - 2*A362389(n-1) for n > 0.

A363580 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^k) * x^k/(k * (1 + 2*x^k)) ).

Original entry on oeis.org

1, 1, 0, 2, 0, 2, 1, 6, -2, 11, -1, 30, -21, 76, -60, 223, -245, 653, -817, 2031, -2935, 6521, -10067, 21455, -35425, 72152, -123756, 246752, -436854, 855852, -1546777, 3001811, -5513604, 10630676, -19747742, 37949424, -71115077, 136415279, -257301742, 493313335

Offset: 0

Seiichi Manyama, Jun 10 2023

sign

Cf. A052855, A198518, A363545, A363546, A363581.

Cf. A363578.

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

A(x) = (1 + 2*x) * B(x) where B(x) is the g.f. of A363578.

a(n) = A363578(n) + 2*A363578(n-1) for n > 0.

A363581 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^k) * x^k/(k * (1 + 3*x^k)) ).

Original entry on oeis.org

1, 1, -1, 4, -6, 11, -22, 62, -151, 353, -867, 2261, -5861, 15178, -39878, 106099, -283823, 763248, -2065453, 5621318, -15368682, 42190539, -116281176, 321647511, -892617214, 2484583934, -6935203356, 19408586888, -54447145335, 153084848495

Offset: 0

Seiichi Manyama, Jun 10 2023

sign

Cf. A052855, A198518, A363545, A363546, A363580.

Cf. A363579.

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

A(x) = (1 + 3*x) * B(x) where B(x) is the g.f. of A363579.

a(n) = A363579(n) + 3*A363579(n-1) for n > 0.

A363548 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^k) * x^k/(k * (1 - x^k)^3) ).

Original entry on oeis.org

1, 1, 5, 19, 79, 326, 1414, 6198, 27794, 126233, 580885, 2700135, 12665756, 59869222, 284919675, 1364009722, 6564545500, 31742029545, 154134718727, 751316355122, 3674923035139, 18031965040197, 88734141475113, 437813286219942, 2165445447313147

Offset: 0

Seiichi Manyama, Jun 09 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A052855, A363547.

Cf. A363507, A363546.

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

A(x) = (1 - x)^3 * B(x) where B(x) is the g.f. of A363507.

a(n) = Sum_{k=0..3} (-1)^k * binomial(3,k) * A363507(n-k).

cp's OEIS Frontend

A363541 G.f. satisfies A(x) = exp( Sum_{k>=1} (3^k + A(x^k)) * x^k/k ).

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A363545 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^k) * x^k/(k * (1 - 2*x^k)) ).

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A363580 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^k) * x^k/(k * (1 + 2*x^k)) ).

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A363581 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^k) * x^k/(k * (1 + 3*x^k)) ).

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A363548 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^k) * x^k/(k * (1 - x^k)^3) ).

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