|
%I M1790 #34 Apr 21 2024 21:08:36
%S 1,2,7,32,178,1160,8653,72704,679798,7005632,78939430,965988224,
%T 12762344596,181108102016,2748049240573,44405958742016,
%U 761423731533286,13809530704348160
%N a(n+1) = (n+1)*a(n) + Sum a(k)*a(n-k).
%D D. E. Knuth, personal communication.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Michael De Vlieger, <a href="/A006014/b006014.txt">Table of n, a(n) for n = 1..449</a>
%H Jimmy Devillet and Bruno Teheux, <a href="https://arxiv.org/abs/1805.11936">Associative, idempotent, symmetric, and order-preserving operations on chains</a>, arXiv:1805.11936 [math.RA], 2018.
%H E. Duchi, V. Guerrini, S. Rinaldi, and G. Schaeffer, <a href="https://doi.org/10.1088/1751-8121/50/2/024002">Fighting fish</a>. J. Phys. A, Math. Theor. 50, No. 2, Article ID 024002, 16 p. (2017), chapter 4.
%F G.f. A(x) satisfies A(x) = x * (1 + A(x) + A(x)^2 + x * A'(x)). - _Michael Somos_, Jul 24 2011
%F a(n) = Sum_{k=0..2^(n-1) - 1} b(k) for n > 0 where b(2n+1) = b(n), b(2n) = b(n) + b(n - 2^f(n)) + b(2n - 2^f(n)) + b(A025480(n-1)) for n > 0 with b(0) = b(1) = 1 and where f(n) = A007814(n). - _Mikhail Kurkov_, Nov 19 2021 [verification needed]
%e x + 2*x^2 + 7*x^3 + 32*x^4 + 178*x^5 + 1160*x^6 + 8653*x^7 + 72704*x^8 + ...
%t Nest[Append[#1, #1[[-1]] (#2 + 1) + Total@ Table[#1[[k]] #1[[#2 - k]], {k, #2 - 1}]] & @@ {#, Length@ #} &, {1}, 17] (* _Michael De Vlieger_, Aug 22 2018 *)
%t (* or *)
%t a[1] = 1; a[n_] := a[n] = n a[n-1] + Sum[a[k] a[n-1-k], {k, n-2}]; Array[a, 18] (* _Giovanni Resta_, Aug 23 2018 *)
%o (PARI) {a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = k * A[k-1] + sum( j=1, k-2, A[j] * A[k-1-j])); A[n])} /* _Michael Somos_, Jul 24 2011 */
%Y Similar recurrences: A124758, A243499, A284005, A329369, A341392.
%K nonn,easy,changed
%O 1,2
%A _N. J. A. Sloane_
|
