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A286180 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k in powers of x. 9
1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 1, 0, 1, 4, 3, 2, 0, 0, 1, 5, 6, 4, 2, 0, 0, 1, 6, 10, 8, 6, 0, 1, 0, 1, 7, 15, 15, 13, 3, 3, 0, 0, 1, 8, 21, 26, 25, 12, 6, 2, 0, 0, 1, 9, 28, 42, 45, 31, 14, 9, 0, 0, 0, 1, 10, 36, 64, 77, 66, 35, 24, 3, 2, 1, 0, 1, 11, 45 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

A(n, k) is the number of ways of writing n as the sum of k triangular numbers.

LINKS

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

FORMULA

G.f. of column k: (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k.

EXAMPLE

Square array begins:

   1, 1, 1, 1,  1,  1, ...

   0, 1, 2, 3,  4,  5, ...

   0, 0, 1, 3,  6, 10, ...

   0, 1, 2, 4,  8, 15, ...

   0, 0, 2, 6, 13, 25, ...

MATHEMATICA

Table[Function[k, SeriesCoefficient[Product[(1 + x^i) (1 - x^(2 i)), {i, Infinity}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten (* Michael De Vlieger, May 07 2017 *)

CROSSREFS

Columns k=0-12 give A000007, A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787.

Main diagonal gives A106337.

Sequence in context: A275001 A290975 A291678 * A291701 A286352 A175045

Adjacent sequences:  A286177 A286178 A286179 * A286181 A286182 A286183

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, May 07 2017

STATUS

approved

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Last modified February 17 23:47 EST 2018. Contains 299297 sequences. (Running on oeis4.)