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 A290429 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j*(j+1)*(j+2)/6))^k. 3
 1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 1, 0, 1, 5, 6, 1, 2, 0, 0, 1, 6, 10, 4, 3, 2, 0, 0, 1, 7, 15, 10, 5, 6, 0, 0, 0, 1, 8, 21, 20, 10, 12, 3, 0, 0, 0, 1, 9, 28, 35, 21, 21, 12, 0, 1, 0, 0, 1, 10, 36, 56, 42, 36, 30, 4, 3, 0, 1, 0, 1, 11, 45, 84, 78, 63, 61, 20, 6, 3, 2, 0, 0, 1, 12, 55, 120, 135, 112, 112, 60, 15, 12, 3, 2, 0, 0 (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 a sum of k tetrahedral (or triangular pyramidal) numbers (A000292). LINKS Seiichi Manyama, Antidiagonals n = 0..139, flattened Eric Weisstein's World of Mathematics, Tetrahedral Number FORMULA G.f. of column k: (Sum_{j>=0} x^(j*(j+1)*(j+2)/6))^k. EXAMPLE Square array begins: 1,  1,  1,  1,   1,   1,  ... 0,  1,  2,  3,   4,   5,  ... 0,  0,  1,  3,   6,  10,  ... 0,  0,  0,  1,   4,  10,  ... 0,  1,  2,  3,   5,  10,  ... 0,  0,  2,  6,  12,  21,  ... MATHEMATICA Table[Function[k, SeriesCoefficient[Sum[x^(i (i + 1) (i + 2)/6), {i, 0, n}]^k, {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten CROSSREFS Cf. A000292, A104246, A286180, A290054, A290430. Cf. A000007 (column 0), A023533 (column 1), A282172 (column 5). Main diagonal gives A303170. Similar to, but different from, A045847. Sequence in context: A052553 A290054 A290430 * A045847 A137586 A291170 Adjacent sequences:  A290426 A290427 A290428 * A290430 A290431 A290432 KEYWORD nonn,tabl AUTHOR Ilya Gutkovskiy, Jul 31 2017 STATUS approved

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Last modified January 20 08:18 EST 2020. Contains 331081 sequences. (Running on oeis4.)