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 A177379 E.g.f. satisfies: A(x) = 1/(1-x - x*log(A(x))). 1
 1, 1, 4, 27, 260, 3270, 50904, 946134, 20462896, 505137312, 14020517520, 432340670520, 14667108820704, 542979374426736, 21784934875431168, 941691211940974320, 43634507604383543040, 2157698329617806488320 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS FORMULA E.g.f.: A(x) = 1/(1 - Series_Reversion(x/(1 - log(1-x)))). ... Let G(x) = e.g.f. of A138013, then: . A(x) = exp(G(x) - 1), . A(x) = 1/(1 - x*G(x)) where G(x) = 1 - log(1 - x*G(x)). ... Let F(x) = e.g.f. of A177380, then: . [x^n] A(x)^(-n+1)/(-n+1) = A177380(n)/n! for n>1, . [x^n] F(x)^(n+1)/(n+1) = a(n)/n! for n>=0, . A(x) = F(x*A(x)) and A(x/F(x)) = F(x), . A(x) = (1/x)*Series_Reversion(x/F(x)) where F(x) = 1+x + x*log(F(x)). Contribution from Paul D. Hanna, Jul 16 2010: (Start) E.g.f. satisfies: A(x) = (1 + x*A'(x)/A(x))*(1 - x*A(x))/(1-x). ... Let A_n(x) denote the n-th iteration of x*A(x) with G = x/(1-x), then: . A(x) = 1 + G + G*Dx(G)/2! + G*Dx(G*Dx(G))/3! + G*Dx(G*Dx(G*Dx(G)))/4! +... . A_n(x)/x = 1 + n*G + n^2*G*Dx(G)/2! + n^3*G*Dx(G*Dx(G))/3! + n^4*G*Dx(G*Dx(G*Dx(G)))/4! +... where Dx(F) = d/dx(x*F). ... Given e.g.f. A(x), the matrix log of the Riordan array (A(x),x*A(x)) equals the matrix L defined by L(n,k)=k+1 and L(n,n)=0, for n>=0, n>k. (End) a(n) ~ sqrt(s-1) * n^(n-1) * s^(n+1) / exp(n), where s = -LambertW(-1,-exp(-2)) = 3.14619322062... (see A226572). - Vaclav Kotesovec, Jan 11 2014 EXAMPLE E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 260*x^4/4! +... Log(A(x)) = G(x) - 1 where G(x) = e.g.f. of A138013 begins: G(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 146*x^4/4! + 1694*x^5/5! + ... and satisfies: exp(1 - G(x)) = 1 - x*G(x). Contribution from Paul D. Hanna, Jul 16 2010: (Start) Given e.g.f. A(x), and A179424 = Riordan array (A(x),x*A(x)) where the g.f. of column k in A179424 equals A(x)^(k+1): 1; 1, 1; 4/2!, 2, 1; 27/3!, 10/2!, 3, 1; 260/4!, 78/3!, 18/2!, 4, 1; 3270/5!, 832/4!, 159/3!, 28/2!, 5, 1; ... then the matrix log of A179424 equals the triangular matrix: 0; 1, 0; 1, 2, 0; 1, 2, 3, 0; 1, 2, 3, 4, 0; 1, 2, 3, 4, 5, 0; ... in which column k equals k+1 in row n for n>k>=0. (End) MATHEMATICA CoefficientList[1/(1-InverseSeries[Series[x/(1-Log[1-x]), {x, 0, 20}], x]), x]*Range[0, 20]! (* Vaclav Kotesovec, Jan 11 2014 *) PROG (PARI) {a(n)=n!*polcoeff(1/(1-serreverse(x/(1-log(1-x+x*O(x^n))))), n)} Contribution from Paul D. Hanna, Jul 16 2010: (Start) (PARI) /* Using matrix log of Riordan array (A(x), x*A(x)): */ {a(n)=local(L=matrix(n+1, n+1, r, c, if(r>c, c)), M=sum(m=0, #L, L^m/m!)); n!*M[n+1, 1]} (PARI) /* From A(x) = (1 + x*A'(x)/A(x))*(1 - x*A(x))/(1-x): */ {a(n)=local(A=1+x); for(k=2, n, A=A-polcoeff((1+x*deriv(A)/A)*(1-x*A)/(1-x+x*O(x^n)), k)*x^k/(k-1)); n!*polcoeff(A, n)} (End) CROSSREFS Cf. A177380, A138013, A226572. Cf. A179424. [From Paul D. Hanna, Jul 16 2010] Sequence in context: A086783 A050764 A240582 * A052813 A218653 A121353 Adjacent sequences:  A177376 A177377 A177378 * A177380 A177381 A177382 KEYWORD nonn AUTHOR Paul D. Hanna, May 15 2010 STATUS approved

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