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%I #25 Jun 12 2021 23:21:26
%S 1,1,1,6,1,2,7,6,2,9,78,7,12,9,24,420,78,14,54,24,130,6872,420,156,63,
%T 144,130,720,17253,6872,840,702,168,780,720,8505,326552,17253,13744,
%U 3780,1872,910,4320,8505,35840
%N Triangle read by rows, T(n,k) = (A156791(n-k+1) * (A006973 * 0^(n-k))).
%C This is a convolution triangle: A006973 convolved with A156791.
%C As a property of eigentriangles, sum of n-th row terms = rightmost term of next row.
%H G. C. Greubel, <a href="/A156792/b156792.txt">Rows n = 0..50 of the triangle, flattened</a>
%F Triangle read by rows, T(n,k) = M*Q =(A156791(n-k+1) * (A006973 * 0^(n-k)))
%F M = an infinite lower triangular matrix with A156791: (1, 1, 6, 7, 78, ...) in every column.
%F Q = an infinite lower triangular matrix with A006973 prefaced with a 1 as the main diagonal: (1, 1, 2, 9, 24, 130, 720, 8505, ...) and the rest zeros.
%F Sum_{k=0..n} T(n, k) = A006973(n+1).
%F T(n, 0) = T(n+1, 1) = A156791(n).
%F T(n, k) = A156791(n-k+2)*([k=0] + A006973(k+1)). - _G. C. Greubel_, Jun 11 2021
%e First few rows of the triangle:
%e 1,
%e 1, 1;
%e 6, 1, 2;
%e 7, 6, 2, 9;
%e 78, 7, 12, 9, 24;
%e 420, 78, 14 54, 24, 130;
%e 6872, 420, 156, 63, 144, 130, 720;
%e 17253, 6872, 840, 702, 168, 780, 720, 8505;
%e 326552, 17253, 13744, 3780, 1872, 910, 4320, 8505, 35840;
%e ...
%e Example: Row 4 = (7, 6, 2, 9) = termwise products of (7, 6, 1, 1) and (1, 1, 2, 9).
%t A006973[n_]:= A006973[n]= If[n<4, Max[n-1, 0], (n-1)!*(1 + Sum[k*(-A006973[k]/k!)^(n/k), {k, Most[Divisors[n]]}])];
%t S[n_, x_]:= Sum[A006973[j]*x^j, {j, 0, n+2}];
%t A156791:= With[{p=100}, CoefficientList[Series[S[p,x]/(x + S[p,x]), {x,0,p}], x]]
%t A156792[n_, k_]:= A156791[[n-k+2]]*(Boole[k==0] + A006973[k+1]);
%t Table[A156792[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 11 2021 *)
%Y Cf. A006973, A156791.
%K eigen,nonn,tabl
%O 0,4
%A _Gary W. Adamson_, Feb 15 2009
%E Typo in last line of triangle corrected by _Olivier Gérard_, Aug 11 2016
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