This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A306548 #83 Oct 22 2022 17:09:06
%S A306548 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,
%T A306548 118,32,1,0,0,7,56,259,560,418,64,1,0,0,8,84,560,2003,3104,1510,128,1,
%U A306548 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
%N 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.
%C A306548 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.
%H A306548 D. V. Widder et al., <a href="https://doi.org/10.1090/S0002-9904-1954-09828-2">The Convolution Transform</a>, Bull. Amer. Math. Soc. 60 (1954), 444-456.
%H A306548 Wikipedia, <a href="https://en.wikipedia.org/wiki/Convolution">Convolution</a>.
%H A306548 Wikipedia, <a href="https://en.wikipedia.org/wiki/Convolution_power">Convolution power</a>.
%F A306548 f(m, s) = s^m, if s >= 0;
%F A306548 f(m, s) = 0, otherwise.
%F A306548 F(n,m) = Sum_{k} f(m, n-k) * f(m, k), -oo < k < +oo;
%F A306548 T(n,k) = F(n-k, k).
%e A306548 ==================================================================
%e A306548 k= 0 1 2 3 4 5 6 7 8 9 10
%e A306548 ==================================================================
%e A306548 n=0: 2;
%e A306548 n=1: 2, 0;
%e A306548 n=2: 3, 0, 0;
%e A306548 n=3: 4, 1, 0, 0;
%e A306548 n=4: 5, 4, 1, 0, 0;
%e A306548 n=5: 6, 10, 8, 1, 0, 0;
%e A306548 n=6: 7, 20, 34, 16, 1, 0, 0;
%e A306548 n=7: 8, 35, 104, 118, 32, 1, 0, 0;
%e A306548 n=8: 9, 56, 259, 560, 418, 64, 1, 0, 0;
%e A306548 n=9: 10, 84, 560, 2003, 3104, 1510, 128, 1, 0, 0;
%e A306548 n=10: 11, 120, 1092, 5888, 16003, 17600, 5554, 256, 1, 0; 0;
%e A306548 ...
%t A306548 f[m_, s_] := Piecewise[{{s^m, s >= 0}, {0, True}}];
%t A306548 F[n_, m_] := Sum[f[m, n - k]*f[m, k], {k, -Infinity, +Infinity}];
%t A306548 T[n_, k_] := F[n - k, k];
%t A306548 Column[Table[T[n, k], {n, 0, 12}, {k, 0, n}], Left]
%Y A306548 Nonzero terms of columns k=0..5 give: A000027, A000292, A033455, A145216, A145217, A145218.
%Y A306548 Partial sums of columns k=1..2 give: A000332, A259181.
%Y A306548 Cf. A302971, A304042, A198633, A000079.
%K A306548 nonn,tabl
%O A306548 0,7
%A A306548 _Kolosov Petro_, Feb 23 2019
%E A306548 Edited by _Kolosov Petro_, Mar 13 2019