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A143918
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G.f. A(x) satisfies: A(x) = 1/(1-x)^2 + x^2*A'(x).
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1
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1, 2, 5, 14, 47, 194, 977, 5870, 41099, 328802, 2959229, 29592302, 325515335, 3906184034, 50780392457, 710925494414, 10663882416227, 170622118659650, 2900576017214069, 52210368309853262, 991996997887211999, 19839939957744240002, 416638739112629040065
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = 3*floor(e*(n-1)!) - 1, n>1. - Gary Detlefs, Jun 10 2010
a(n) = (n-1) * a(n-1) + n + 1 for n > 0 and a(0) = 1. - Werner Schulte, Oct 20 2023
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 5*x^2 + 14*x^3 + 47*x^4 + 194*x^5 + 977*x^6 +...
x^2*A'(x) = 2*x^2 + 10*x^3 + 42*x^4 + 188*x^5 + 970*x^6 + 5862*x^7 +...
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MAPLE
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a:= proc(n) option remember; `if`(n<3, n^2+1,
((n^2+1)*a(n-1)-(n-2)*(n+1)*a(n-2))/n)
end:
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MATHEMATICA
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a[0]=1; a[n_]:=(n-1)*a[n-1]+n+1; Array[a, 23, 0] (* Stefano Spezia, Oct 20 2023 *)
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PROG
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(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x+x*O(x^n))^2+x^2*deriv(A)); polcoeff(A, n)}
(PARI) A143918_upto(N)=vector(N, n, N=if(n>1, (n-2)*N+n, 1)) \\ Gives the N initial values a(0..N-1). - M. F. Hasler, Oct 20 2023
(PARI) A143918(n)=if(n>1, localprec(max(logint(n=(n-1)!, 10), 5)+5); n\exp(-1)*3-1, n+1) \\ M. F. Hasler, Oct 20 2023
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CROSSREFS
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Cf. A000522 (floor(e*n!) for n > 0).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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