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

0,3

Links

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

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.

Example

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;
  ...

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 *)

Prog

(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(""))
(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<x>:=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
(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.<x> = 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

Crossrefs

Cf. A084784, A084785, A084786.

Sequence in context: A278298 A178927 A076571 * A265853 A129838 A032153

Adjacent sequences: A084780 A084781 A084782 * A084784 A084785 A084786

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 13 2003

Status

approved