A293377 -id:A293377 - cp's OEIS frontend

A290216 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} jx^(ji)).

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 2, 0, 1, 1, 3, 2, 2, 0, 1, 1, 3, 5, 6, 3, 0, 1, 1, 3, 5, 6, 7, 4, 0, 1, 1, 3, 5, 10, 10, 12, 5, 0, 1, 1, 3, 5, 10, 10, 18, 13, 6, 0, 1, 1, 3, 5, 10, 15, 22, 25, 22, 8, 0, 1, 1, 3, 5, 10, 15, 22, 29, 34, 26, 10, 0, 1, 1, 3, 5

Offset: 0

Seiichi Manyama, Oct 06 2017

			Square array begins:
   1, 1, 1,  1,  1, ...
   0, 1, 1,  1,  1, ...
   0, 1, 3,  3,  3, ...
   0, 2, 2,  5,  5, ...
   0, 2, 6,  6, 10, ...
   0, 3, 7, 10, 10, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A000007, A000009, A293204, A290269.
Rows n=0 gives A000012.
Main diagonal gives A077285.
Product_{i>0} (1 + Sum_{j=1..k} j*x^(j*i))^m: A290217 (m=-1), this sequence (m=1), A293377 (m=2).
Cf. A293305.

A293378 Expansion of (eta(q^6)/(eta(q)eta(q^2)eta(q^3)))^2 in powers of q.

Original entry on oeis.org

1, 2, 7, 16, 39, 80, 171, 328, 638, 1168, 2133, 3744, 6540, 11092, 18687, 30816, 50421, 81136, 129582, 204160, 319340, 493952, 758781, 1154624, 1745748, 2617958, 3902614, 5776144, 8501784, 12434320, 18092565, 26175784, 37689734, 53989056, 76993497, 109284736

Offset: 0

Seiichi Manyama, Oct 07 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A077285, A293377.

nmax = 50; CoefficientList[Series[Product[((1 + x^(3*k))/((1 - x^k)*(1 - x^(2*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 11 2017 *)

G.f.: Product_{k>0} ((1 - x^(6*k))/((1 - x^k)*(1 - x^(2*k))*(1 - x^(3*k))))^2.

a(n) ~ 5^(5/4) * exp(2*Pi*sqrt(5*n)/3) / (72 * sqrt(3) * n^(7/4)). - Vaclav Kotesovec, Oct 11 2017

A293386 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} jx^(ji))^2.

Original entry on oeis.org

1, 1, 0, 1, -2, 0, 1, -2, 1, 0, 1, -2, -3, -2, 0, 1, -2, -3, 10, 4, 0, 1, -2, -3, 4, -4, -4, 0, 1, -2, -3, 4, 14, -20, 5, 0, 1, -2, -3, 4, 6, -8, 41, -6, 0, 1, -2, -3, 4, 6, 16, -46, 2, 9, 0, 1, -2, -3, 4, 6, 6, -30, 14, -111, -12, 0, 1, -2, -3, 4, 6, 6, 0, -58

Offset: 0

Seiichi Manyama, Oct 07 2017

			Square array begins:
   1,  1,   1,  1,  1, ...
   0, -2,  -2, -2, -2, ...
   0,  1,  -3, -3, -3, ...
   0, -2,  10,  4,  4, ...
   0,  4,  -4, 14,  6, ...
   0, -4, -20, -8, 16, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..1 give A000007, A022597.
Rows n=0 gives A000012.
Main diagonal gives A252650.
Product_{i>0} (1 + Sum_{j=1..k} j*x^(j*i))^m: this sequence (m=-2), A290217 (m=-1), A290216 (m=1), A293377 (m=2).

cp's OEIS Frontend

A290216 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} jx^(ji)).

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A293378 Expansion of (eta(q^6)/(eta(q)eta(q^2)eta(q^3)))^2 in powers of q.

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A293386 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} jx^(ji))^2.

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cp's OEIS Frontend

A290216 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} j*x^(j*i)).

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A293378 Expansion of (eta(q^6)/(eta(q)*eta(q^2)*eta(q^3)))^2 in powers of q.

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Programs

Mathematica

Formula

A293386 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} j*x^(j*i))^2.

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A290216 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} jx^(ji)).

A293378 Expansion of (eta(q^6)/(eta(q)eta(q^2)eta(q^3)))^2 in powers of q.

A293386 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(1 + Sum_{j=1..k} jx^(ji))^2.