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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.
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%I #21 Apr 19 2018 16:26:10

%S 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,

%T 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,

%U 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

%N 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.

%C A(n,k) is the number of ways of writing n as a sum of k tetrahedral (or triangular pyramidal) numbers (A000292).

%H Seiichi Manyama, <a href="/A290429/b290429.txt">Antidiagonals n = 0..139, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TetrahedralNumber.html">Tetrahedral Number</a>

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>

%F G.f. of column k: (Sum_{j>=0} x^(j*(j+1)*(j+2)/6))^k.

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, ...

%e 0, 1, 2, 3, 4, 5, ...

%e 0, 0, 1, 3, 6, 10, ...

%e 0, 0, 0, 1, 4, 10, ...

%e 0, 1, 2, 3, 5, 10, ...

%e 0, 0, 2, 6, 12, 21, ...

%t 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

%Y Cf. A000292, A104246, A286180, A290054, A290430.

%Y Cf. A000007 (column 0), A023533 (column 1), A282172 (column 5).

%Y Main diagonal gives A303170.

%Y Similar to, but different from, A045847.

%K nonn,tabl

%O 0,8

%A _Ilya Gutkovskiy_, Jul 31 2017