A307495 Expansion of Sum_{k>=0} k!*((1 - sqrt(1 - 4*x))/2)^k.

1, 1, 3, 12, 57, 312, 1950, 13848, 111069, 998064, 9957186, 109305240, 1309637274, 17006109072, 237888664572, 3566114897520, 57030565449765, 969154436550240, 17439499379433690, 331268545604793240, 6624013560942038670, 139080391965533653200, 3059323407592802838180, 70355685298375014175440

Offset: 0

Author

Ilya Gutkovskiy, Apr 10 2019

Keywords

• nonn

Comments

Catalan transform of A000142 (factorial numbers).

From Peter Bala, Jan 27 2020: (Start)

This sequence is the main diagonal of the lower triangular array formed by putting the sequence of factorial numbers in the first column (k = 0) of the array and then completing the triangle using the relation T(n,k) = T(n-1,k) + T(n,k-1) for k >= 1.

1

1 1

2 3 3

6 9 12 12

24 33 45 57 57

120 153 198 255 312 312

...

Alternatively, the sequence can be obtained by multiplying the sequence of factorial numbers by the array A106566.

(End)

Links

• P. Bala, A note on the Catalan transform of a sequence

Crossrefs

Cf. A000108, A000142, A013999, A100100, A106566, A307496.

Programs

Mathematica

nmax = 23; CoefficientList[Series[Sum[k! ((1 - Sqrt[1 - 4 x])/2)^k, {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[1/(1 + ContinuedFractionK[-Floor[(k + 1)/2] (1 - Sqrt[1 - 4 x])/2, 1, {k, 1, nmax}]), {x, 0, nmax}], x]
Join[{1}, Table[1/n Sum[Binomial[2n - k - 1, n - k] k k!, {k, n}], {n, 23}]]

Formula

G.f.: 1 /(1 - x*c(x)/(1 - x*c(x)/(1 - 2*x*c(x)/(1 - 2*x*c(x)/(1 - 3*x*c(x)/(1 - 3*x*c(x)/(1 - ...))))))), a continued fraction, where c(x) = g.f. of Catalan numbers (A000108).

Sum_{n>=0} a(n)*(x*(1 - x))^n = g.f. of A000142.

a(n) = (1/n) * Sum_{k=1..n} binomial(2*n-k-1,n-k)*k*k! for n > 0.

a(n) ~ exp(1) * n!. - Vaclav Kotesovec, Aug 10 2019