A303172 -id:A303172 - cp's OEIS frontend

A303170 Number of ordered ways of writing n as a sum of n tetrahedral numbers.

1, 1, 1, 1, 5, 21, 61, 141, 309, 757, 2111, 6051, 16721, 44617, 118301, 318501, 871781, 2400741, 6596953, 18067329, 49460555, 135697395, 373271515, 1028451579, 2835353337, 7819016521, 21572619771, 59562583471, 164586609409, 455114644297, 1259191262441, 3485551053561

Offset: 0

Ilya Gutkovskiy, Apr 19 2018

nonn

Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers

Main diagonal of A290429.

Cf. A000292, A068980, A106337, A282582, A303172.

Table[SeriesCoefficient[Sum[x^(k (k + 1) (k + 2)/6), {k, 0, n}]^n, {x, 0, n}], {n, 0, 31}]

a(n) = [x^n] (Sum_{k>=0} x^(k*(k+1)*(k+2)/6))^n.

a(n) = A290429(n,n).

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.

Original entry on oeis.org

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, 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, 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

Offset: 0

Ilya Gutkovskiy, Jul 31 2017

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

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

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Square Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000330, A045847, A122141, A286815, A290429.
Cf. A000007 (column 0), A253903 (column 1), A282173 (column 6).
Main diagonal gives A303172.

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

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

cp's OEIS Frontend

A303170 Number of ordered ways of writing n as a sum of n tetrahedral numbers.

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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.

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cp's OEIS Frontend

A303170 Number of ordered ways of writing n as a sum of n tetrahedral numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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.