A155866 A 'Morgan Voyce' transform of the Bell numbers A000110.

1, 2, 6, 22, 91, 413, 2032, 10754, 60832, 365815, 2327835, 15612872, 109992442, 811500784, 6253327841, 50211976959, 419239644142, 3632891419054, 32616077413970, 302915722319509, 2906047810600157, 28761123170398258, 293296874302640254, 3078390856651377534, 33220524976632438215

Offset: 0

Author

Paul Barry, Jan 29 2009

Keywords

• easy
• nonn

Comments

Image of Bell numbers under Riordan array (1/(1-x), x/(1-x)^2).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..550
• Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.

Crossrefs

Cf. A000110, A085478.

Programs

Magma

[(&+[Binomial(n+j,2*j)*Bell(j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 10 2021

Mathematica

A155866[n_]:= Sum[Binomial[n+j, 2*j]*BellB[j], {j,0,n}];
Table[A155866[n], {n, 0, 30}] (* G. C. Greubel, Jun 10 2021 *)

Sage

def A155866(n): return sum( binomial(n+j, 2*j)*bell_number(j) for j in (0..n) )
[A155866(n) for n in (0..30)] # G. C. Greubel, Jun 10 2021

Formula

G.f.: 1/(1 -x -x/(1 -x -x/(1 -x -x/(1 -x -2*x/(1 -x -x/(1 -x -3*x/(1 -x -x/(1 -x -4*x/(1 - ... (continued fraction).

a(n) = Sum_{k=0..n} binomial(n+k,2k)*A000110(k).

a(n) = Sum_{k=0..n} A085478(n,k)*A000110(k). - Philippe Deléham, Jan 31 2009