A001996 -id:A001996 - cp's OEIS frontend

A001994 Expansion of 1/((1-x^2)(1-x^4)^2(1-x^6)(1-x^8)(1-x^10)) (even powers only).

1, 1, 3, 4, 8, 11, 18, 24, 36, 47, 66, 84, 113, 141, 183, 225, 284, 344, 425, 508, 617, 729, 872, 1020, 1205, 1397, 1632, 1877, 2172, 2480, 2846, 3228, 3677, 4146, 4691, 5261, 5917, 6603, 7386, 8205, 9133, 10103, 11195, 12336, 13613, 14947, 16431, 17981, 19697

Offset: 0

N. J. A. Sloane

A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (9, -33, 63, -66, 36, -8).

Cf. A001996.

nn = 202; t = CoefficientList[Series[1/((1 - x^2)*(1 - x^4)^2*(1 - x^6)*(1 - x^8)*(1 - x^10)), {x, 0, nn}], x]; t = Take[t, {1, nn, 2}]

More terms from James Sellers, Feb 09 2000

A333925 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j=2..k+1} 1/(1 - x^j).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 2, 1, 2, 0, 0, 1, 0, 1, 1, 2, 2, 3, 1, 1, 0, 1, 0, 1, 1, 2, 2, 3, 2, 2, 0, 0, 1, 0, 1, 1, 2, 2, 4, 3, 4, 2, 1, 0, 1, 0, 1, 1, 2, 2, 4, 3, 5, 3, 2, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 6, 5, 5, 2, 1, 0

Offset: 0

Ilya Gutkovskiy, Apr 10 2020

A(n,k) is the number of partitions of n into parts 2, 3, ..., k and k + 1.

			Square array begins:
  1,  1,  1,  1,  1,  1,  ...
  0,  0,  0,  0,  0,  0,  ...
  0,  1,  1,  1,  1,  1,  ...
  0,  0,  1,  1,  1,  1,  ...
  0,  1,  1,  2,  2,  2,  ...
  0,  0,  1,  1,  2,  2,  ...

David A. Corneth, Table of n, a(n) for n = 0..10010 (first 141 rows antidiagonals flattened)
Index entries for Molien series

Columns k=0..12 give A000007, A059841, A103221, A266755, A008667, A037145, A001996, A266776, A266777, A266778, A266779, A266780, A266781.
Main diagonal gives A002865.
Cf. A008284, A058398.

Table[Function[k, SeriesCoefficient[Product[1/(1 - x^j), {j, 2, k + 1}], {x, 0, n}]][i - n], {i, 0, 13}, {n, 0, i}] // Flatten

G.f. of column k: Product_{j=2..k+1} 1/(1 - x^j).

cp's OEIS Frontend

A001994 Expansion of 1/((1-x^2)(1-x^4)^2(1-x^6)(1-x^8)(1-x^10)) (even powers only).

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A333925 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j=2..k+1} 1/(1 - x^j).

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cp's OEIS Frontend

A001994 Expansion of 1/((1-x^2)*(1-x^4)^2*(1-x^6)*(1-x^8)*(1-x^10)) (even powers only).

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A333925 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j=2..k+1} 1/(1 - x^j).

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Formula

A001994 Expansion of 1/((1-x^2)(1-x^4)^2(1-x^6)(1-x^8)(1-x^10)) (even powers only).