A363509 -id:A363509 - cp's OEIS frontend

A363507 G.f. satisfies A(x) = exp( Sum_{k>=1} (3 + A(x^k)) * x^k/k ).

1, 4, 14, 50, 191, 763, 3180, 13640, 59937, 268304, 1219626, 5614038, 26117296, 122598622, 579977691, 2762264225, 13234003724, 63737225733, 308406648979, 1498558628584, 7309116199687, 35772044402485, 175621484712091, 864670723348447

Offset: 0

Seiichi Manyama, Jun 06 2023

nonn

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

Cf. A000081, A029857, A036249, A363508.

Cf. A363509.

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

A(x) = Sum_{k>=0} a(k) * x^k = 1/(1-x)^3 * 1/Product_{k>=0} (1-x^(k+1))^a(k).

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( 3 + Sum_{d|k} d * a(d-1) ) * a(n-k).

A363510 G.f. satisfies A(x) = exp( Sum_{k>=1} (-1)^(k+1) * (4 + A(x^k)) * x^k/k ).

Original entry on oeis.org

1, 5, 15, 50, 190, 766, 3231, 14066, 62681, 284591, 1311622, 6120183, 28855529, 137257541, 657894518, 3174411715, 15406640415, 75162477018, 368383443235, 1813007892858, 8956214966017, 44393932344984, 220732441125743, 1100621484436502

Offset: 0

Seiichi Manyama, Jun 06 2023

nonn

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

Cf. A004111, A038075, A038076, A363509.

Cf. A363508.

seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, (-1)^(k+1)*(4+subst(A, x, x^k))*x^k/k)+x*O(x^n))); Vec(A);

A(x) = Sum_{k>=0} a(k) * x^k = (1+x)^4 * Product_{k>=0} (1+x^(k+1))^a(k).

a(0) = 1; a(n) = (-1/n) * Sum_{k=1..n} ( 4 * (-1)^k + Sum_{d|k} (-1)^(k/d) * d * a(d-1) ) * a(n-k).

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).

A363566 G.f. satisfies A(x) = exp( Sum_{k>=1} (3 * (-1)^k + A(x^k)) * x^k/k ).

Original entry on oeis.org

1, -2, 2, 0, -2, 0, 5, -3, -9, 11, 16, -34, -27, 102, 30, -296, 56, 807, -548, -2056, 2572, 4770, -9846, -9351, 33822, 11496, -107296, 17853, 316498, -210013, -862785, 1069352, 2122294, -4347217, -4402138, 15657617, 5883290, -51677928, 7420844, 157867636

Offset: 0

Seiichi Manyama, Jun 10 2023

sign

Cf. A001678, A363565.

Cf. A363509, A363575.

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

A(x) = B(x)/(1 + x)^3 where B(x) is the g.f. of A363575.
A(x) = Sum_{k>=0} a(k) * x^k = 1/(1+x)^3 * 1/Product_{k>=0} (1-x^(k+1))^a(k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( 3 * (-1)^k + Sum_{d|k} d * a(d-1) ) * a(n-k).

cp's OEIS Frontend

A363507 G.f. satisfies A(x) = exp( Sum_{k>=1} (3 + A(x^k)) * x^k/k ).

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A363510 G.f. satisfies A(x) = exp( Sum_{k>=1} (-1)^(k+1) * (4 + A(x^k)) * x^k/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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A363566 G.f. satisfies A(x) = exp( Sum_{k>=1} (3 * (-1)^k + A(x^k)) * x^k/k ).

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