OFFSET
1,2
REFERENCES
D. E. Knuth, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..449
Jimmy Devillet and Bruno Teheux, Associative, idempotent, symmetric, and order-preserving operations on chains, arXiv:1805.11936 [math.RA], 2018.
E. Duchi, V. Guerrini, S. Rinaldi, and G. Schaeffer, Fighting fish. J. Phys. A, Math. Theor. 50, No. 2, Article ID 024002, 16 p. (2017), chapter 4.
FORMULA
G.f. A(x) satisfies A(x) = x * (1 + A(x) + A(x)^2 + x * A'(x)). - Michael Somos, Jul 24 2011
Conjecture: 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
a(n) ~ cosh(sqrt(3)*Pi/2) * n! / Pi = A073017 * n!. - Vaclav Kotesovec, Nov 21 2024
EXAMPLE
x + 2*x^2 + 7*x^3 + 32*x^4 + 178*x^5 + 1160*x^6 + 8653*x^7 + 72704*x^8 + ...
MATHEMATICA
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 *)
(* or *)
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 *)
PROG
(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 */
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
STATUS
approved