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 A179330 E.g.f. satisfies: A(x) = (1+x)/(1+3*x) * A(x*(1+x)^2). 2
 0, 2, -6, 42, -468, 7080, -133128, 2938824, -73169568, 1997384832, -58814501760, 1868053207680, -65311214042880, 2585560450337280, -115344597684718080, 5424254194395456000, -244310147229735014400, 10256126830544041574400 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS FORMULA E.g.f. A=A(x) satisfies: . (1+x)^2 = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! +... . (1+x)^2*(1+x*(1+x)^2)^2 = 1 + 2*A + 2^2*A*Dx(A)/2! + 2^3*A*Dx(A*Dx(A))/3! + 2^4*A*Dx(A*Dx(A*Dx(A)))/4! +... . G001764(-x)^2 = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+...; G001764(x) = g.f. of A001764; where Dx(F) = d/dx(x*F). INVERSION FORMULA: More generally, if A(x) = A(G(x)) * G(x)/(x*G'(x)) with G(0)=0, G'(0)=1, then G(x) can be obtained from A=A(x) by the series: . G(x)/x = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! +... where Dx(F) = d/dx(x*F). ITERATION FORMULA: Let G_n(x) denote the n-th iteration of G(x) = x*(1+x)^2, and A=A(x), then: . A(x) = A(G_n(x)) * G_n(x)/(x*G_n'(x)) for all n; . G_n(x)/x = 1 + n*A + n^2*A*Dx(A)/2! + n^3*A*Dx(A*Dx(A))/3! + n^4*A*Dx(A*Dx(A*Dx(A)))/4! +... where Dx(F) = d/dx(x*F). ... MATRIX LOG OF RIORDAN ARRAY (G(x)/x, G(x)) where G(x) = x*(1+x)^2: . k*A(x) = e.g.f. of column k of the matrix log of triangle A116088 for k>=0. EXAMPLE E.g.f.: A(x) = 2*x - 6*x^2/2! + 42*x^3/3! - 468*x^4/4! + 7080*x^5/5! - 133128*x^6/6! + 2938824*x^7/7! - 73169568*x^8/8! + 1997384832*x^9/9! - 58814501760*x^10/10! + 1868053207680*x^11/11! - 65311214042880*x^12/12! +... ... A(x*(1+x)^2) = 2*x + 2*x^2/2! - 18*x^3/3! + 108*x^4/4! - 480*x^5/5! - 2808*x^6/6! + 162792*x^7/7! - 3940128*x^8/8! + 57267648*x^9/9! + 534366720*x^10/10! - 78703384320*x^11/11! + 2883142045440*x^12/12! +... ... where A(x*(1+x)^2) = (1+3*x)/(1+x) * A(x). ... Related expansions begin: . A = 2*x - 6*x^2/2! + 42*x^3/3! - 468*x^4/4! + 7080*x^5/5! +... . A*Dx(A)/2! = 8*x^2/2! - 90*x^3/3! + 1332*x^4/4! - 25200*x^5/5! +... . A*Dx(A*Dx(A))/3! = 48*x^3/3! - 1248*x^4/4! + 32760*x^5/5! -+... . A*Dx(A*Dx(A*Dx(A)))/4! = 384*x^4/4! - 18480*x^5/5! + 770400*x^6/6! -+... . A*Dx(A*Dx(A*Dx(A*Dx(A))))/5! = 3840*x^5/5! - 300672*x^6/6! +-... ... Sums of which generate the square of the g.f. of A001764: . G001764(-x)^2 = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+... . G001764(-x)^2 = 1 - 2*x + 7*x^2 - 30*x^3 + 143*x^4 - 728*x^5 +...+ A006013(n)*(-x)^n +... ... The Riordan array ((1+x)^2, x*(1+x)^2) (cf. A116088) begins: 1; 2, 1; 1, 4, 1; 0, 6, 6, 1; 0, 4, 15, 8, 1; 0, 1, 20, 28, 10, 1; 0, 0, 15, 56, 45, 12, 1; ... The matrix log of Riordan array ((1+x)^2, x*(1+x)^2) begins: 0; 2, 0; -6/2!, 4, 0; 42/3!, -12/2!, 6, 0; -468/4!, 84/3!, -18/2!, 8, 0; 7080/5!, -936/4!, 126/3!, -24/2!, 10, 0; -133128/6!, 14160/5!, -1404/4!, 168/3!, -30/2!, 12, 0; ... where the g.f. of the leftmost column equals the e.g.f. of this sequence. PROG (PARI) /* E.g.f. satisfies: A(x) = (1+x)/(1+3*x)*A(x*(1+x)^2): */ {a(n)=local(A=2*x, B); for(m=2, n, B=(1+x)/(1+3*x+O(x^(n+3)))*subst(A, x, x*(1+x)^2+O(x^(n+3))); A=A-polcoeff(B, m+1)*x^m/(m-1)/2); n!*polcoeff(A, n)} (PARI) /* (1+x)^2 = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! +...: */ {a(n)=local(A=0+sum(m=1, n-1, a(m)*x^m/m!), D=1, R=0); R=-((1+x)^2+x*O(x^n))+1+sum(m=1, n, (D=A*deriv(x*D+x*O(x^n)))/m!); -n!*polcoeff(R, n)} (PARI) /* First column of the matrix log of triangle A116088: */ {a(n)=local(M=matrix(n+1, n+1, r, c, if(r>=c, polcoeff(((1+x)^2+x*O(x^n))^c, r-c))), LOG, ID=M^0); LOG=sum(m=1, n+1, -(ID-M)^m/m); n!*LOG[n+1, 1]} CROSSREFS Cf. A179331, variants: A179320, A179420. Sequence in context: A258969 A161632 A115974 * A066864 A181737 A116896 Adjacent sequences:  A179327 A179328 A179329 * A179331 A179332 A179333 KEYWORD sign AUTHOR Paul D. Hanna, Jul 21 2010 STATUS approved

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Last modified January 16 17:49 EST 2022. Contains 350376 sequences. (Running on oeis4.)