cp's OEIS Frontend

A006014 a(n+1) = (n+1)*a(n) + Sum_{k=1..n-1} a(k)*a(n-k).

%I A006014 M1790 #44 Jan 10 2025 20:42:55
%S A006014 1,2,7,32,178,1160,8653,72704,679798,7005632,78939430,965988224,
%T A006014 12762344596,181108102016,2748049240573,44405958742016,
%U A006014 761423731533286,13809530704348160,264141249701936818,5314419112717217792,112201740111374359516,2480395343617443024896
%N A006014 a(n+1) = (n+1)*a(n) + Sum_{k=1..n-1} a(k)*a(n-k).
%D A006014 D. E. Knuth, personal communication.
%D A006014 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006014 Michael De Vlieger, <a href="/A006014/b006014.txt">Table of n, a(n) for n = 1..449</a>
%H A006014 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 A006014 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 A006014 G.f. A(x) satisfies A(x) = x * (1 + A(x) + A(x)^2 + x * A'(x)). - _Michael Somos_, Jul 24 2011
%F A006014 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
%F A006014 a(n) ~ cosh(sqrt(3)*Pi/2) * n! / Pi = A073017 * n!. - _Vaclav Kotesovec_, Nov 21 2024
%e A006014 G.f. = x + 2*x^2 + 7*x^3 + 32*x^4 + 178*x^5 + 1160*x^6 + 8653*x^7 + 72704*x^8 + ...
%t A006014 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 A006014 (* or *)
%t A006014 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 A006014 (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 */
%o A006014 (Python)
%o A006014 from sage.all import *
%o A006014 @CachedFunction
%o A006014 def a(n): # a = A006014
%o A006014     if n<5: return (pow(5,n-1) + 3)//4
%o A006014     else: return n*a(n-1) + 2*sum(a(k)*a(n-k-1) for k in range(1,(n//2))) + (n%2)*pow(a((n-1)//2),2)
%o A006014 print([a(n) for n in range(1,61)]) # _G. C. Greubel_, Jan 10 2025
%Y A006014 Similar recurrences: A124758, A243499, A284005, A329369, A341392.
%Y A006014 Cf. A007814, A025480, A073017.
%K A006014 nonn,easy
%O A006014 1,2
%A A006014 _N. J. A. Sloane_