A084786 Row sums of the triangle (A084783) and the differences of the main diagonal (A084785) and the first column (A084784).

1, 3, 10, 41, 211, 1354, 10620, 99327, 1081744, 13443065, 187538132, 2899087774, 49149083790, 906169148064, 18044322039456, 385825735367745, 8814867042465387, 214270073007359704, 5520898403200292418, 150290771692227728963, 4309813692713979537503

Offset: 0

Author

Paul D. Hanna, Jun 13 2003

Keywords

• nonn

Comments

In the triangle (A084783), the diagonal (A084785) is the self-convolution of the first column (A084784) and the row sums (this sequence) gives the differences of the diagonal and the first column.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A084783, A084784, A084785.

Programs

Mathematica

A084784= With[{m=60}, CoefficientList[Series[Exp[Sum[Sum[ j!*StirlingS2[k, j], {j, k}]*x^k /k , {k, m + 1}]], {x,0,m}], x]];
T[n_, k_]:= T[n, k]= If[k==0, A084784[[n+1]], T[n, k-1] + T[n-1, k-1]]; (* A084783 *)
A084786[n_]:= A084786[n]= Sum[T[n, k], {k,0,n}];
Table[A084786[n], {n,0,40}] (* G. C. Greubel, Jun 08 2023 *)

PARI

A = matrix(25, 25); A[1, 1] = 1; rs = 1; print(1); for (n = 2, 25, sc = sum (i = 2, n - 1, A[i, 1]*A[n + 1 - i, 1]); A[n, 1] = rs - sc; rs = A[n, 1]; for (k = 2, n, A[n, k] = A[n, k - 1] + A[n - 1, k - 1]; rs += A[n, k]); print(rs)); \\ David Wasserman, Jan 06 2005

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)
@CachedFunction
def T(n,k): # T = A084783
    if k==0: return b[n]
    else: return T(n, k-1) + T(n-1, k-1)
def A084786(n): return sum(T(n, k) for k in range(n+1))
[A084786(n) for n in range(m-9)] # G. C. Greubel, Jun 08 2023

Formula

a(n) ~ n! / (2 * (log(2))^(n+2)). - Vaclav Kotesovec, Nov 19 2014

Extensions

More terms from David Wasserman, Jan 06 2005