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.

1, 1, 2, 2, 3, 5, 6, 8, 11, 16, 25, 31, 39, 50, 66, 137, 162, 193, 232, 282, 348, 944, 1081, 1243, 1436, 1668, 1950, 2298, 7884, 8828, 9909, 11152, 12588, 14256, 16206, 18504, 77514, 85398, 94226, 104135, 115287, 127875, 142131, 158337, 176841

Offset: 0

Author

Paul D. Hanna, Jun 13 2003

Keywords

• nonn
• tabl

Examples

			Triangle begins:
      1;
      1,     2;
      2,     3,     5;
      6,     8,    11,     16;
     25,    31,    39,     50,     66;
    137,   162,   193,    232,    282,    348;
    944,  1081,  1243,   1436,   1668,   1950,   2298;
   7884,  8828,  9909,  11152,  12588,  14256,  16206,  18504;
  77514, 85398, 94226, 104135, 115287, 127875, 142131, 158337, 176841;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..150, flattened (first 45 rows from Paul D. Hanna)

Crossrefs

Cf. A084784, A084785, A084786.

Programs

Magma

m:=50;
f:= func< n,x | Exp((&+[(&+[Factorial(j)*StirlingSecond(k,j)*x^k/k: j in [1..k]]): k in [1..n+2]])) >;
R:=PowerSeriesRing(Rationals(), m+1);
b:=Coefficients(R!( f(m,x) )); // b = A084784
function T(n,k) // T = A084783
  if k eq 0 then return b[n+1];
  else return T(n,k-1) + T(n-1,k-1);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jun 08 2023

Maple

T:= proc(n, k) option remember; `if`(k=0, 1+add(T(j, 0)*
     (binomial(n, j)-T(n-j, 0)), j=1..n-1), T(n, k-1)+T(n-1, k-1))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 09 2023

Mathematica

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[n_, k_]:= T[n, k]= If[k==0, b[n], T[n, k-1] +T[n-1, k-1]];
Table[T[n,k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 07 2023 *)

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 */
{T(n,k)=if(k==0,if(n==0,1,A084784(n)),T(n, k-1)+T(n-1, k-1))}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

SageMath

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)))
m=50
def A084784_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(m,x) ).list()
b=A084784_list(m)
def T(n,k): # T = A084783
    if k==0: return b[n]
    else: return T(n, k-1) + T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Jun 08 2023

Formula

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.