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A084786
Row sums of the triangle (A084783) and the differences of the main diagonal (A084785) and the first column (A084784).
4
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
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 13 2003
EXTENSIONS
More terms from David Wasserman, Jan 06 2005
STATUS
approved