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A290430 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)*(2*j+1)/6))^k. 3

%I

%S 1,1,0,1,1,0,1,2,0,0,1,3,1,0,0,1,4,3,0,0,0,1,5,6,1,0,1,0,1,6,10,4,0,2,

%T 0,0,1,7,15,10,1,3,2,0,0,1,8,21,20,5,4,6,0,0,0,1,9,28,35,15,6,12,3,0,

%U 0,0,1,10,36,56,35,12,20,12,0,0,0,0,1,11,45,84,70,28,31,30,4,0,1,0,0,1,12,55,120,126,64,49,60,20,0,3,0,0,0

%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)*(2*j+1)/6))^k.

%C A(n,k) is the number of ways of writing n as a sum of k square pyramidal numbers (A000330).

%H Seiichi Manyama, <a href="/A290430/b290430.txt">Antidiagonals n = 0..139, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquarePyramidalNumber.html">Square Pyramidal 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)*(2*j+1)/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, 0, 0, 0, 1, 5, ...

%e 0, 1, 2, 3, 4, 6, ...

%t Table[Function[k, SeriesCoefficient[Sum[x^(i (i + 1) (2 i + 1)/6), {i, 0, n}]^k, {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

%Y Cf. A000330, A045847, A122141, A286815, A290429.

%Y Cf. A000007 (column 0), A253903 (column 1), A282173 (column 6).

%Y Main diagonal gives A303172.

%K nonn,tabl

%O 0,8

%A _Ilya Gutkovskiy_, Jul 31 2017

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Last modified February 23 06:13 EST 2020. Contains 332159 sequences. (Running on oeis4.)