A363541 -id:A363541 - cp's OEIS frontend

A362389 G.f. satisfies A(x) = exp( Sum_{k>=1} (2^k + A(x^k)) * x^k/k ).

1, 3, 10, 34, 122, 450, 1723, 6758, 27135, 110913, 460395, 1935233, 8222504, 35255000, 152353021, 662892684, 2901595559, 12768195617, 56450822365, 250637657015, 1117060889815, 4995815027658, 22413020866875, 100842092305575, 454912716037387

Offset: 0

Seiichi Manyama, Jun 09 2023

nonn

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

Cf. A001678, A036249, A363541.

Cf. A029857, A363542, A363545.

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 A363545.
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).

A363546 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, 5, 22, 105, 497, 2431, 11976, 59928, 302816, 1545660, 7955132, 41255625, 215378364, 1131134574, 5972272636, 31684600709, 168824599282, 903080385252, 4848038120323, 26110774945462, 141048622038068, 764026532321068, 4149020129689451

Offset: 0

Seiichi Manyama, Jun 09 2023

nonn

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

Cf. A052855, A198518, A363545, A363580, A363581.

Cf. A363541.

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 A363541.

a(n) = A363541(n) - 3*A363541(n-1) for n > 0.

A363543 G.f. satisfies A(x) = exp( Sum_{k>=1} (-1)^(k+1) * (3^k + A(x^k)) * x^k/k ).

Original entry on oeis.org

1, 4, 7, 23, 69, 234, 826, 3000, 11168, 42313, 162829, 634052, 2495051, 9903761, 39612048, 159481988, 645833656, 2628829700, 10749777653, 44139474552, 181916530895, 752288709592, 3120574260606, 12981015704961, 54138655342763, 226330448292140

Offset: 0

Seiichi Manyama, Jun 09 2023

nonn

Cf. A038075, A363542.

Cf. A363509, A363541.

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

A362389 G.f. satisfies A(x) = exp( Sum_{k>=1} (2^k + A(x^k)) * x^k/k ).

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A363546 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^k) * x^k/(k * (1 - 3*x^k)) ).

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A363543 G.f. satisfies A(x) = exp( Sum_{k>=1} (-1)^(k+1) * (3^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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