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

0,2

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

Formula

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

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 *)

Prog

(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.<x> = 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

Crossrefs

Cf. A084783, A084784, A084785.

Sequence in context: A030942 A030855 A030954 * A156170 A245502 A009329

Adjacent sequences: A084783 A084784 A084785 * A084787 A084788 A084789

Keyword

nonn

Author

Paul D. Hanna, Jun 13 2003

Extensions

More terms from David Wasserman, Jan 06 2005

Status

approved