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A006014
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a(n+1) = (n+1)*a(n) + Sum a(k)*a(n-k).
(Formerly M1790)
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5
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1, 2, 7, 32, 178, 1160, 8653, 72704, 679798, 7005632, 78939430, 965988224, 12762344596, 181108102016, 2748049240573, 44405958742016, 761423731533286, 13809530704348160
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OFFSET
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1,2
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REFERENCES
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D. E. Knuth, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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.
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FORMULA
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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
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EXAMPLE
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x + 2*x^2 + 7*x^3 + 32*x^4 + 178*x^5 + 1160*x^6 + 8653*x^7 + 72704*x^8 + ...
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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STATUS
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approved
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