A293463 -id:A293463 - cp's OEIS frontend

A293489 E.g.f.: Product_{m>0} (1 + x^(2m-1) + x^(4m-2)/2! + x^(6*m-3)/3!).

1, 1, 1, 7, 24, 180, 1200, 10080, 97440, 997920, 12096000, 149688000, 2128896000, 31135104000, 515804889600, 8717829120000, 162151621632000, 3092504583168000, 63934815200256000, 1366817864315904000, 31019500604252160000, 733013100880220160000

Offset: 0

Seiichi Manyama, Oct 10 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..446

Column k=3 of A293486.

Cf. A293463.

N=66; x='x+O('x^N); Vec(serlaplace(prod(m=1, N, 1+x^(2*m-1)+x^(4*m-2)/2+x^(6*m-3)/6)))

A293461 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^(j(2*i-1))).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 4, 1, 1, 0, 1, 1, 2, 4, 1, 3, 1, 0, 1, 1, 2, 4, 5, 3, 3, 1, 0, 1, 1, 2, 4, 5, 3, 6, 5, 2, 0, 1, 1, 2, 4, 5, 8, 6, 5, 6, 2, 0, 1, 1, 2, 4, 5, 8, 6, 9, 9, 4, 2, 0, 1, 1, 2, 4, 5, 8, 12, 9, 9, 13

Offset: 0

Seiichi Manyama, Oct 09 2017

			Square array begins:
   1, 1, 1, 1, 1, ...
   0, 1, 1, 1, 1, ...
   0, 0, 2, 2, 2, ...
   0, 1, 1, 4, 4, ...
   0, 1, 1, 1, 5, ...
   0, 1, 3, 3, 3, ...

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

Columns k=0..3 give A000007, A000700, A293304, A293463.
Rows n=0..1 give A000012, A057427.
Main diagonal gives A102186.
Cf. A290216.

max = 12; A[n_, k_] := SeriesCoefficient[Product[(x*(-(k*x^((2*i - 1)*(k + 1) + 1)) - x^((2*i - 1)*(k + 1) + 1) + k*x^((2*i - 1)*(k + 1) + 2*i) + x^(2*i)))/(x^(2*i) - x)^2 + 1, {i, 1, max}], {x, 0, n}]; Flatten[ Table[ A[n - k, k], {n, 0, max}, {k, n, 0, -1}]] (* Jean-François Alcover, Oct 10 2017 *)

cp's OEIS Frontend

A293489 E.g.f.: Product_{m>0} (1 + x^(2m-1) + x^(4m-2)/2! + x^(6*m-3)/3!).

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A293461 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^(j(2*i-1))).

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cp's OEIS Frontend

A293489 E.g.f.: Product_{m>0} (1 + x^(2*m-1) + x^(4*m-2)/2! + x^(6*m-3)/3!).

Views

Author

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Programs

PARI

A293461 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*(2*i-1))).

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Mathematica

A293489 E.g.f.: Product_{m>0} (1 + x^(2m-1) + x^(4m-2)/2! + x^(6*m-3)/3!).

A293461 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^(j(2*i-1))).