|
%I #13 Nov 23 2021 09:45:41
%S 1,2,10,77,798,10392,162996,2991340,62893270,1490758022,39334017996,
%T 1143492521437,36318168041260,1251270023475864,46481870133666792,
%U 1852054390616046345,78792796381529620710,3564894013016856836190,170921756533520140861020,8657018996674423681277455,461881087606113071895396420
%N a(n) = n! * [x^n] exp((n+2)*x)*(BesselI(0,2*x) - BesselI(1,2*x)).
%C The n-th term of the n-th binomial transform of A000108.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F a(n) = [x^n] (sqrt(1 - n*x) - sqrt(1 - 4*x - n*x))/(2*x*sqrt(1 - n*x)).
%F a(n) = A271025(n,n).
%F a(n) ~ exp(2) * (BesselI(0,2) - BesselI(1,2)) * n^n. - _Vaclav Kotesovec_, Sep 20 2017
%F a(n) = Sum_{k=0..n} binomial(n,k) * A000108(k) * n^(n-k). - _Vaclav Kotesovec_, Nov 23 2021
%t 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 *)
%t Join[{1}, Table[Sum[Binomial[n, j] * CatalanNumber[j] * n^(n-j), {j, 0, n}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Nov 23 2021 *)
%Y Main diagonal of A271025.
%Y Cf. A000108, A292631, A349603.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Sep 20 2017
|
