A283536 -id:A283536 - cp's OEIS frontend

A283534 Expansion of exp( Sum_{n>=1} -A283533(n)/n*x^n ) in powers of x.

1, -1, -16, -713, -64687, -9688545, -2165715003, -675843665621, -280752874575386, -149800127959983890, -99844730502381895830, -81300082280849836639246, -79413710313923588156379547, -91652445699847071535357000689, -123383623610527054787988720527285, -191626051373071219208574650313032502

Offset: 0

Seiichi Manyama, Mar 10 2017

sign

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

Cf. Product_{k>=1} (1 - x^k)^(k^(m*k)): A010815 (m=0), A283499 (m=1), this sequence (m=2), A283536 (m=3).

Cf. A283579 (Product_{k>=1} 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) = if(n==0, 1, -(1/n)*sum(k=1, n, sumdiv(k, d, d^(2*d + 1))*a(n - k)));
for(n=0, 15, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 11 2017

G.f.: Product_{k>=1} (1 - x^k)^(k^(2*k)).

a(n) = -(1/n)*Sum_{k=1..n} A283533(k)*a(n-k) for n > 0.

A283580 Expansion of exp( Sum_{n>=1} A283535(n)/n*x^n ) in powers of x.

Original entry on oeis.org

1, 1, 65, 19748, 16799044, 30535636881, 101591759812967, 558649739234980142, 4722932373908389412037, 58154498193439779564557624, 1000058469893323150011227885608, 23226158305362748824532880463596385, 708825166389400019044145225134521489486

Offset: 0

Seiichi Manyama, Mar 11 2017

nonn

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

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

Cf. A283536 (Product_{k>=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/(1 - x^k)^(k^(3*k)).
a(n) = (1/n)*Sum_{k=1..n} A283535(k)*a(n-k) for n > 0.
a(n) ~ n^(3*n) * (1 + exp(-3)/n^3). - Vaclav Kotesovec, Mar 17 2017

A283675 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-x^j)^(j^(k*j)) in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -1, -1, 1, -1, -4, 0, 1, -1, -16, -23, 0, 1, -1, -64, -713, -223, 1, 1, -1, -256, -19619, -64687, -2767, 0, 1, -1, -1024, -531185, -16755517, -9688545, -42268, 1, 1, -1, -4096, -14347883, -4294403215, -30499543213, -2165715003, -759008, 0, 1, -1, -16384

Offset: 0

Seiichi Manyama, Mar 14 2017

			Square array begins:
   1,    1,      1,         1, ...
  -1,   -1,     -1,        -1, ...
  -1,   -4,    -16,       -64, ...
   0,  -23,   -713,    -19619, ...
   0, -223, -64687, -16755517, ...

Seiichi Manyama, Antidiagonals n = 0..52, flattened

Columns k=0..4 give A010815, A283499, A283534, A283536, A283803.
Rows n=0..1 give A000012, (-1)*A000012.
Main diagonal gives A283720.
Cf. A283674.

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

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(k*d+1)) * A(n-j,k) for n > 0. - Seiichi Manyama, Nov 04 2017

A283803 Expansion of exp( Sum_{n>=1} -A283369(n)/n*x^n ) in powers of x.

Original entry on oeis.org

1, -1, -256, -531185, -4294403215, -95363000657073, -4738284730302658391, -459981771468075494207385, -79227701254823507875355278590, -22528320196093613328344381426130010, -9999977451048811940735941180766259658078

Offset: 0

Seiichi Manyama, Mar 17 2017

sign

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

Cf. Product_{k>=1} (1 - x^k)^(k^(m*k)): A010815 (m=0), A283499 (m=1), A283534 (m=2), A283536 (m=3), this sequence (m=4).

Cf. A283510 (Product_{k>=1} 1/(1 - x^k)^(k^(4*k))).

CoefficientList[Series[Product[(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 - x^k)^(k^(4*k)).

a(n) = -(1/n)*Sum_{k=1..n} A283369(k)*a(n-k) for n > 0.

cp's OEIS Frontend

A283534 Expansion of exp( Sum_{n>=1} -A283533(n)/n*x^n ) in powers of x.

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A283580 Expansion of exp( Sum_{n>=1} A283535(n)/n*x^n ) in powers of x.

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A283675 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-x^j)^(j^(k*j)) in powers of x.

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A283803 Expansion of exp( Sum_{n>=1} -A283369(n)/n*x^n ) in powers of x.

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Mathematica

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