OFFSET
0,3
LINKS
Alois P. Heinz, Rows n = 0..150, flattened (first 45 rows from Paul D. Hanna)
FORMULA
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jun 13 2003
STATUS
approved