A362389 -id:A362389 - 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.

A363542 G.f. satisfies A(x) = exp( Sum_{k>=1} (-1)^(k+1) * (2^k + A(x^k)) * x^k/k ).

Original entry on oeis.org

1, 3, 5, 14, 38, 114, 360, 1166, 3872, 13094, 44961, 156244, 548636, 1943333, 6935817, 24917586, 90039163, 327029681, 1193258619, 4371901789, 16077606949, 59325057056, 219579151797, 815017718383, 3032959638204, 11313632991360, 42295634914403

Offset: 0

Seiichi Manyama, Jun 09 2023

nonn

Cf. A038075, A363543.

Cf. A362389.

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

A(x) = Sum_{k>=0} a(k) * x^k = (1+2*x) * 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} (-1)^(k/d) * d * a(d-1) ) * a(n-k).

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

Original entry on oeis.org

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).

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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A363542 G.f. satisfies A(x) = exp( Sum_{k>=1} (-1)^(k+1) * (2^k + A(x^k)) * x^k/k ).

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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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