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 A179421 E.g.f. A(x) satisfies: x*A(x) equals column 0 in the matrix log of the Riordan array (A(x), x*A(x)). 7
 1, 1, 4, 33, 440, 8380, 211824, 6771422, 264621216, 12329282160, 671882721600, 42181858413552, 3013915118776704, 242626985772839616, 21821596448977248000, 2176989083049432207600, 239420370429753669425664 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..150 FORMULA a(n) = A179420(n+1)/(n+1). E.g.f. A(x) satisfies: A(x) = x*A'(x)/(A(x*A(x)) - 1). Let L_{n}(x) denote the n-th iteration of L=L(x)=x*A(x) then: . L(L(x)) = x*L'(x), . L_{n+1}(x) = L(x)*x*[d/dx L_n(x)]/L_n(x), . L(x)/x = 1 + L + L*Dx(L)/2! + L*Dx(L*Dx(L))/3! + L*Dx(L*Dx(L*Dx(L)))/4! +... . L_{n}(x)/x = 1 + n*L + n^2*L*Dx(L)/2! + n^3*L*Dx(L*Dx(L))/3! + n^4*L*Dx(L*Dx(L*Dx(L)))/4! +... where Dx(F) = d/dx(x*F). EXAMPLE E.g.f.: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 440*x^4/4! +... x*A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2200*x^5/5! +... ... The RIORDAN ARRAY (A(x), x*A(x)) begins: . 1; . 1, 1; . 4/2!, 2, 1; . 33/3!, 10/2!, 3, 1; . 440/4!, 90/3!, 18/2!, 4, 1; . 8380/5!, 1240/4!, 177/3!, 28/2!, 5, 1; . 211824/6!, 23800/5!, 2544/4!, 300/3!, 40/2!, 6, 1; . 6771422/7!, 598788/6!, 49680/5!, 4520/4!, 465/3!, 54/2!, 7, 1; ... where the e.g.f. of column k = A(x)^(k+1) for k>=0. ... The MATRIX LOG of the above Riordan array (A(x), x*A(x)) begins: . 0; . 1, 0; . 2/2!, 2, 0; . 12/3!, 4/2!, 3, 0; . 132/4!, 24/3!, 6/2!, 4, 0; . 2200/5!, 264/4!, 36/3!, 8/2!, 5, 0; . 50280/6!, 4400/5!, 396/4!, 48/3!, 10/2!, 6, 0; . 1482768/7!, 100560/6!, 6600/5!, 528/4!, 60/3!, 12/2!, 7, 0; ... where the e.g.f. of column k = (k+1)*x*A(x) for k>=0. PROG (PARI) {a(n)=local(A=1+x+sum(m=2, n-1, a(m)*x^m/m!)+x*O(x^n)); if(n<2, n!*polcoeff(A, n), n!*polcoeff(subst(x*A, x, x*A)/x, n)/(n-1))} CROSSREFS Cf. A179420, A179422, A179423, A179424. Sequence in context: A102321 A268293 A193421 * A296835 A331690 A002190 Adjacent sequences:  A179418 A179419 A179420 * A179422 A179423 A179424 KEYWORD nonn AUTHOR Paul D. Hanna, Jul 15 2010 STATUS approved

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Last modified April 14 12:11 EDT 2021. Contains 342949 sequences. (Running on oeis4.)