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A104988 Matrix square of triangle A104980.

%I A104988 #9 Jun 07 2021 09:27:10
%S A104988 1,2,1,8,4,1,42,20,6,1,266,120,38,8,1,1954,836,270,62,10,1,16270,6616,
%T A104988 2150,516,92,12,1,151218,58576,19030,4688,882,128,14,1,1551334,573672,
%U A104988 185674,46516,9050,1392,170,16,1,17414114,6159976,1982310,502324,99994,15956,2070,218,18,1
%N A104988 Matrix square of triangle A104980.
%C A104988 Triangular matrix A104980 satisfies: SHIFT_LEFT(column 0 of A104980^p) = p*(column p+1 of A104980) for p>=0.
%H A104988 G. C. Greubel, <a href="/A104988/b104988.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A104988 T(n+1, 0) = 2*A104980(n+3, 3) = 2*A104982(n) for n>=0.
%e A104988 Triangle begins:
%e A104988          1;
%e A104988          2,       1;
%e A104988          8,       4,       1;
%e A104988         42,      20,       6,      1;
%e A104988        266,     120,      38,      8,     1;
%e A104988       1954,     836,     270,     62,    10,     1;
%e A104988      16270,    6616,    2150,    516,    92,    12,    1;
%e A104988     151218,   58576,   19030,   4688,   882,   128,   14,   1;
%e A104988    1551334,  573672,  185674,  46516,  9050,  1392,  170,  16,  1;
%e A104988   17414114, 6159976, 1982310, 502324, 99994, 15956, 2070, 218, 18, 1;
%t A104988 t[n_, k_]:= t[n, k]= If[k<0 || k>n, 0, If[k==n, 1, If[k==n-1, n, k*t[n, k+1] + Sum[t[j, 0]*t[n, j+k+1], {j, 0, n-k-1}]]]]; (* t = A104980 *)
%t A104988 M:= With[{q=20}, Table[If[j>i, 0, t[i, j]], {i,0,q}, {j,0,q}]];
%t A104988 Table[MatrixPower[M, 2][[n+1, k+1]], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 07 2021 *)
%o A104988 (PARI) T(n,k)= if(n<k || k<0, 0, (matrix(n+1, n+1, m, j, if(m==j, 1, if(m==j+1, -m+1, -polcoeff((1-1/sum(i=0,m,i!*x^i))/x+O(x^m), m-j-1))))^-2)[n+1,k+1])
%Y A104988 Cf. A104980, A104982 (column 0), A104989 (row sums).
%K A104988 nonn,tabl
%O A104988 0,2
%A A104988 _Paul D. Hanna_, Apr 10 2005