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A084783 Triangle, read by rows, such that the diagonal (A084785) is the self-convolution of the first column (A084784) and the row sums (A084786) gives the differences of the diagonal and the first column.

%I A084783 #21 Jun 09 2023 18:23:46
%S A084783 1,1,2,2,3,5,6,8,11,16,25,31,39,50,66,137,162,193,232,282,348,944,
%T A084783 1081,1243,1436,1668,1950,2298,7884,8828,9909,11152,12588,14256,16206,
%U A084783 18504,77514,85398,94226,104135,115287,127875,142131,158337,176841
%N A084783 Triangle, read by rows, such that the diagonal (A084785) is the self-convolution of the first column (A084784) and the row sums (A084786) gives the differences of the diagonal and the first column.
%H A084783 Alois P. Heinz, <a href="/A084783/b084783.txt">Rows n = 0..150, flattened</a> (first 45 rows from Paul D. Hanna)
%F A084783 T(0,0) = 1, T(n,0) = A084784(n), T(n,n) = A084785(n), T(n,k) = T(n,k-1) + T(n-1,k-1) for n>0, k>0.
%e A084783 Triangle begins:
%e A084783       1;
%e A084783       1,     2;
%e A084783       2,     3,     5;
%e A084783       6,     8,    11,     16;
%e A084783      25,    31,    39,     50,     66;
%e A084783     137,   162,   193,    232,    282,    348;
%e A084783     944,  1081,  1243,   1436,   1668,   1950,   2298;
%e A084783    7884,  8828,  9909,  11152,  12588,  14256,  16206,  18504;
%e A084783   77514, 85398, 94226, 104135, 115287, 127875, 142131, 158337, 176841;
%e A084783   ...
%p A084783 T:= proc(n, k) option remember; `if`(k=0, 1+add(T(j, 0)*
%p A084783      (binomial(n, j)-T(n-j, 0)), j=1..n-1), T(n, k-1)+T(n-1, k-1))
%p A084783     end:
%p A084783 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Jun 09 2023
%t A084783 b[n_]:= b[n]= If[n<1, Boole[n==0], Module[{A= 1/x -1/x^2}, Do[A=2A - Normal@Series[(x A^2)/. x-> x-1, {x, Infinity, k+1}], {k,2,n}]; (-1)^n Coefficient[A, x, -n-1]]]; (* b = A084784 *)
%t A084783 T[n_, k_]:= T[n, k]= If[k==0, b[n], T[n, k-1] +T[n-1, k-1]];
%t A084783 Table[T[n,k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 07 2023 *)
%o A084783 (PARI) {A084784(n) = local(A); if( n<0, 0, A=1; for(k=1, n, A = truncate(A + O(x^k)) + x * O(x^k); A += A - 1 / subst(A^-2, x, x /(1 + x)) / (1 + x); ); polcoeff(A, n))}; /* After Michael Somos */
%o A084783 {T(n,k)=if(k==0,if(n==0,1,A084784(n)),T(n, k-1)+T(n-1, k-1))}
%o A084783 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%o A084783 (Magma)
%o A084783 m:=50;
%o A084783 f:= func< n,x | Exp((&+[(&+[Factorial(j)*StirlingSecond(k,j)*x^k/k: j in [1..k]]): k in [1..n+2]])) >;
%o A084783 R<x>:=PowerSeriesRing(Rationals(), m+1);
%o A084783 b:=Coefficients(R!( f(m,x) )); // b = A084784
%o A084783 function T(n,k) // T = A084783
%o A084783   if k eq 0 then return b[n+1];
%o A084783   else return T(n,k-1) + T(n-1,k-1);
%o A084783   end if;
%o A084783 end function;
%o A084783 [T(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jun 08 2023
%o A084783 (SageMath)
%o A084783 def f(n, x): return exp(sum(sum( factorial(j)*stirling_number2(k,j) *x^k/k for j in range(1,k+1)) for k in range(1,n+2)))
%o A084783 m=50
%o A084783 def A084784_list(prec):
%o A084783     P.<x> = PowerSeriesRing(QQ, prec)
%o A084783     return P( f(m,x) ).list()
%o A084783 b=A084784_list(m)
%o A084783 def T(n,k): # T = A084783
%o A084783     if k==0: return b[n]
%o A084783     else: return T(n, k-1) + T(n-1, k-1)
%o A084783 flatten([[T(n, k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Jun 08 2023
%Y A084783 Cf. A084784, A084785, A084786.
%K A084783 nonn,tabl
%O A084783 0,3
%A A084783 _Paul D. Hanna_, Jun 13 2003