A283580 -id:A283580 - cp's OEIS frontend

A283536 Expansion of exp( Sum_{n>=1} -A283535(n)/n*x^n ) in powers of x.

1, -1, -64, -19619, -16755517, -30499543213, -101528172949440, -558442022082754554, -4721800698082895269442, -58144976385942395405449505, -999941534906642496357956893139, -23224150593200781968944997552887957, -708778584588517237886357058373629079824

Offset: 0

Seiichi Manyama, Mar 10 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..152

Cf. Product_{k>=1} (1 - x^k)^(k^(m*k)): A010815 (m=0), A283499 (m=1), A283534 (m=2), this sequence (m=3).

Cf. A283580 (Product_{k>=1} 1/(1 - x^k)^(k^(3*k))).

A[n_] :=  Sum[d^(3*d + 1), {d, Divisors[n]}]; a[n_]:=If[n==0, 1, -(1/n)*Sum[A[k]*a[n - k], {k, n}]]; Table[a[n], {n, 0, 12}] (* Indranil Ghosh, Mar 11 2017 *)

A(n) = sumdiv(n, d, d^(3*d + 1));
a(n) = if(n==0, 1, -(1/n)*sum(k=1, n, A(k)*a(n - k)));
for(n=0, 12, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 11 2017

G.f.: Product_{k>=1} (1 - x^k)^(k^(3*k)).

a(n) = -(1/n)*Sum_{k=1..n} A283535(k)*a(n-k) for n > 0.

A283579 Expansion of exp( Sum_{n>=1} A283533(n)/n*x^n ) in powers of x.

Original entry on oeis.org

1, 1, 17, 746, 66418, 9843707, 2187941520, 680615139257, 282199700198462, 150389915598653924, 100155578743010743914, 81505577512720707466924, 79580089689432499741178617, 91814299713761739807846854872

Offset: 0

Seiichi Manyama, Mar 11 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..214

Cf. Product_{k>=1} 1/(1 - x^k)^(k^(m*k)): A000041 (m=0), A023880 (m=1), this sequence (m=2), A283580 (m=3).

Cf. A283534 (Product_{k>=1} (1 - x^k)^(k^(2*k))).

A[n_] :=  Sum[d^(2*d + 1), {d, Divisors[n]}]; a[n_] := If[n==0, 1, (1/n)*Sum[A[k]*a[n - k], {k, n}]]; Table[a[n], {n, 0, 13}] (* Indranil Ghosh, Mar 11 2017 *)

A(n) = sumdiv(n, d, d^(2*d + 1));
a(n) = if(n==0, 1, (1/n)*sum(k=1, n, A(k)*a(n - k)));
for(n=0, 11, print1(a(n),", ")) \\ Indranil Ghosh, Mar 11 2017

G.f.: Product_{k>=1} 1/(1 - x^k)^(k^(2*k)).
a(n) = (1/n)*Sum_{k=1..n} A283533(k)*a(n-k) for n > 0.
a(n) ~ n^(2*n) * (1 + exp(-2)/n^2). - Vaclav Kotesovec, Mar 17 2017

A283674 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-x^j)^(j^(k*j)) in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 17, 32, 5, 1, 1, 65, 746, 298, 7, 1, 1, 257, 19748, 66418, 3531, 11, 1, 1, 1025, 531698, 16799044, 9843707, 51609, 15, 1, 1, 4097, 14349932, 4295531890, 30535636881, 2187941520, 894834, 22, 1, 1, 16385, 387424586, 1099526502508, 95371863221411, 101591759812967, 680615139257, 17980052, 30

Offset: 0

Seiichi Manyama, Mar 14 2017

			Square array begins:
   1,   1,     1,        1, ...
   1,   1,     1,        1, ...
   2,   5,    17,       65, ...
   3,  32,   746,    19748, ...
   5, 298, 66418, 16799044, ...

Alois P. Heinz, Antidiagonals n = 0..52

Columns k=0-4 give A000041, A023880, A283579, A283580, A283510.
Rows give: 0-1: A000012, 2: A052539, 3: A283716.
Main diagonal gives A283719.
Cf. A283675.

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
      d*d^(k*d), d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Mar 15 2017

A[n_, k_] := If[n==0, 1, Sum[Sum[d*d^(k*d), {d, Divisors[j]}] *A[n - j, k], {j, n}] / n]; Flatten[Table[A[d - n,  n],{d, 0, 10},{n, d, 0, -1}]] (* Indranil Ghosh, Mar 17 2017 *)

A(n, k) = if(n==0, 1, sum(j=1, n, sumdiv(j, d, d*d^(k*d)) * A(n - j, k))/n);
{for(d=0, 10, for(n=0, d, print1(A(n, d - n),", ");); print(););} \\ Indranil Ghosh, Mar 17 2017

G.f. of column k: Product_{j>=1} 1/(1-x^j)^(j^(k*j)).

A283510 Expansion of exp( Sum_{n>=1} A283369(n)/n*x^n ) in powers of x.

Original entry on oeis.org

1, 1, 257, 531698, 4295531890, 95371863221411, 4738477950914329100, 459991301719292572342573, 79228623778497392212453912974, 22528478894247280128054776211273960, 10000022549030658394108744658459680045250

Offset: 0

Seiichi Manyama, Mar 17 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..120

Cf. Product_{k>=1} 1/(1 - x^k)^(k^(m*k)): A000041 (m=0), A023880 (m=1), A283579 (m=2), A283580 (m=3), this sequence (m=4).

Cf. A283803 (Product_{k>=1} (1 - x^k)^(k^(4*k))).

CoefficientList[Series[Product[1/(1 - x^k)^(k^(4k)), {k, 1, 10}], {x, 0, 10}], x] (* Indranil Ghosh, Mar 17 2017 *)

A(n) = sumdiv(n, d, d^(4*d + 1));
a(n) = if(n<1, 1, (1/n) * sum(k=1, n, A(k) * a(n - k)));
for(n=0, 10, print1(a(n),", ")) \\ Indranil Ghosh, Mar 17 2017

G.f.: Product_{k>=1} 1/(1 - x^k)^(k^(4*k)).
a(n) = (1/n)*Sum_{k=1..n} A283369(k)*a(n-k) for n > 0.
a(n) ~ n^(4*n) * (1 + exp(-4)/n^4). - Vaclav Kotesovec, Mar 17 2017

cp's OEIS Frontend

A283536 Expansion of exp( Sum_{n>=1} -A283535(n)/n*x^n ) in powers of x.

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A283579 Expansion of exp( Sum_{n>=1} A283533(n)/n*x^n ) in powers of x.

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A283674 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-x^j)^(j^(k*j)) in powers of x.

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A283510 Expansion of exp( Sum_{n>=1} A283369(n)/n*x^n ) in powers of x.

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