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