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

0,3

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

Table of n, a(n) for n=0..23.

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

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

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}]]

Crossrefs

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

Sequence in context: A014333 A185618 A027710 * A302101 A279271 A293469

Adjacent sequences: A307492 A307493 A307494 * A307496 A307497 A307498

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 10 2019

Status

approved