A292632 a(n) = n! * [x^n] exp((n+2)*x)*(BesselI(0,2*x) - BesselI(1,2*x)).

1, 2, 10, 77, 798, 10392, 162996, 2991340, 62893270, 1490758022, 39334017996, 1143492521437, 36318168041260, 1251270023475864, 46481870133666792, 1852054390616046345, 78792796381529620710, 3564894013016856836190, 170921756533520140861020, 8657018996674423681277455, 461881087606113071895396420

Offset: 0

Author

Ilya Gutkovskiy, Sep 20 2017

Keywords

• nonn

Comments

The n-th term of the n-th binomial transform of A000108.

Links

• N. J. A. Sloane, Transforms

Crossrefs

Main diagonal of A271025.

Cf. A000108, A292631, A349603.

Programs

Mathematica

Table[n!*SeriesCoefficient[E^((n+2)*x)*(BesselI[0,2*x] - BesselI[1,2*x]),{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 20 2017 *)
Join[{1}, Table[Sum[Binomial[n, j] * CatalanNumber[j] * n^(n-j), {j, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 23 2021 *)

Formula

a(n) = [x^n] (sqrt(1 - n*x) - sqrt(1 - 4*x - n*x))/(2*x*sqrt(1 - n*x)).

a(n) = A271025(n,n).

a(n) ~ exp(2) * (BesselI(0,2) - BesselI(1,2)) * n^n. - Vaclav Kotesovec, Sep 20 2017

a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * n^(n-k). - Vaclav Kotesovec, Nov 23 2021