A306548 Triangle T(n,k) read by rows, where the k-th column is the shifted self-convolution of the power function n^k, n >= 0, 0 <= k <= n.

0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 4, 1, 0, 0, 4, 10, 8, 1, 0, 0, 5, 20, 34, 16, 1, 0, 0, 6, 35, 104, 118, 32, 1, 0, 0, 7, 56, 259, 560, 418, 64, 1, 0, 0, 8, 84, 560, 2003, 3104, 1510, 128, 1, 0, 0, 9, 120, 1092, 5888, 16003, 17600, 5554, 256, 1, 0, 0, 10, 165, 1968, 14988, 64064, 130835, 101504, 20758, 512, 1, 0, 0

Offset: 0

Author

Kolosov Petro, Feb 23 2019

Keywords

• nonn
• tabl

Comments

For n > 0 an odd-power identity n^(2m+1)+1, m >= 0 can be found using the current sequence. The sum of the n-th diagonal of T(n,k) over 0 <= k <= m multiplied by A(m,k) gives n^(2m+1)-1, where A(m,k) = A302971(m,k)/A304042(m,k). For example, consider the case n=4, m=2: the n-th diagonal of T(n, 0 <= k <= m) is {5, 10, 34}, and the m-th row of triangle A(m, 0 <= k <= m) is {1, 0, 30}, thus (3+1)^5 + 1 = 5*1 + 10*0 + 34*30 = 1025.

Examples

			==================================================================
k=    0     1     2     3      4      5     6    7    8    9    10
==================================================================
n=0:  2;
n=1:  2,    0;
n=2:  3,    0,    0;
n=3:  4,    1,    0,    0;
n=4:  5,    4,    1,    0,     0;
n=5:  6,   10,    8,    1,     0,     0;
n=6:  7,   20,   34,   16,     1,     0,    0;
n=7:  8,   35,  104,  118,    32,     1,    0,   0;
n=8:  9,   56,  259,  560,   418,    64,    1,   0,   0;
n=9:  10,  84,  560, 2003,  3104,  1510,  128,   1,   0,   0;
n=10: 11, 120, 1092, 5888, 16003, 17600, 5554, 256,   1,   0;   0;
...
		

Links

• D. V. Widder et al., The Convolution Transform, Bull. Amer. Math. Soc. 60 (1954), 444-456.
• Wikipedia, Convolution.
• Wikipedia, Convolution power.

Crossrefs

Nonzero terms of columns k=0..5 give: A000027, A000292, A033455, A145216, A145217, A145218.

Partial sums of columns k=1..2 give: A000332, A259181.

Cf. A302971, A304042, A198633, A000079.

Programs

Mathematica

f[m_, s_] := Piecewise[{{s^m, s >= 0}, {0, True}}];
F[n_, m_] := Sum[f[m, n - k]*f[m, k], {k, -Infinity, +Infinity}];
T[n_, k_] := F[n - k, k];
Column[Table[T[n, k], {n, 0, 12}, {k, 0, n}], Left]

Formula

f(m, s) = s^m, if s >= 0;

f(m, s) = 0, otherwise.

F(n,m) = Sum_{k} f(m, n-k) * f(m, k), -oo < k < +oo;

T(n,k) = F(n-k, k).

Extensions

Edited by Kolosov Petro, Mar 13 2019