login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

a(0)=a(1)=1; thereafter a(n+1) = 1 + a(n) + Sum_{m=1..n} binomial(2*n, 2*m-1)*a(n+1-m).
1

%I #23 Jan 15 2021 21:10:08

%S 1,1,4,25,262,3991,82132,2173933,71445730,2839498243,133692806800,

%T 7335280745401,462699693768670,33178393683900559,2678741141802447820,

%U 241522439552562161797,24145570512983311489882,2659835689591105375581595,321076697410366882043603848,42263134034904720594569141713

%N a(0)=a(1)=1; thereafter a(n+1) = 1 + a(n) + Sum_{m=1..n} binomial(2*n, 2*m-1)*a(n+1-m).

%H Seiichi Manyama, <a href="/A308008/b308008.txt">Table of n, a(n) for n = 0..296</a>

%H Oleksiy Khorunzhiy, <a href="https://arxiv.org/abs/1904.01339">On asymptotic behavior of Bell polynomials and high moments of vertex degree of random graphs</a>, arXiv:1904.01339 [math.PR], 2019. See (3.18). [Warning: gives wrong value 4001 for a(5).]

%H O. Khorunzhiy, <a href="https://doi.org/10.1007/s10959-020-01025-w">On Asymptotic Properties of Bell Polynomials and Concentration of Vertex Degree of Large Random Graphs</a>, Journal of Theoretical Probability (2020).

%t Nest[Append[#1, 1 + #1[[-1]] + Sum[Binomial[2 #2, 2 m - 1]*#1[[#2 + 2 - m]], {m, #2}]] & @@ {#, Length@ # - 1} &, {1, 1}, 18] (* or *)

%t Block[{a}, a[0] = a[1] = 1; a[n_] := a[n] = 1 + a[n - 1] + Sum[Binomial[2 (n - 1), 2 m - 1] a[n - m], {m, n - 1}]; Array[a[#] &, 20, 0]] (* _Michael De Vlieger_, May 15 2019 *)

%o (PARI) a(n) = if (n<=1, 1, 1 + a(n-1) + sum(m=1, n-1, binomial(2*(n-1), 2*m-1)*a(n-m))); \\ _Michel Marcus_, May 15 2019

%K nonn

%O 0,3

%A _Michael De Vlieger_ and _N. J. A. Sloane_, May 14 2019

%E More terms from _Seiichi Manyama_, May 15 2019