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A135920 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k^2*x).

%I A135920 #47 Apr 25 2024 13:47:54
%S A135920 1,1,2,7,37,264,2433,27913,386906,6346119,121159373,2655174768,
%T A135920 66028903633,1845579100993,57506847262162,1983312152411351,
%U A135920 75238783332550789,3122408658986242072,141063757638078429489
%N A135920 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k^2*x).
%C A135920 From _Peter Bala_, Sep 27 2012: (Start)
%C A135920 Generalized Bell numbers; row sums of A036969.
%C A135920 a(n) is equal to the number of partitions of the set {1,1',2,2',...,n,n'} into disjoint nonempty subsets V1,...,Vk (1 <= k <= n) such that, for each 1 <= j <= k, if i is the least integer such that either i or i' belongs to Vj then {i,i'} is a subset of Vj.
%C A135920 Example: a(3) = 7: There is a single partition into one set {1,1',2,2',3,3'}; five partitions into two sets, namely, {1,1',2,2'}{3,3'}, {1,1',3,3'}{2,2'}, {1,1'}{2,2',3,3'}, {1,1',3}{2,2',3'} and {1,1',3'}{2,2',3}; and finally a single partition into three sets {1,1'}{2,2'}{3,3'}. (End)
%H A135920 S. Matsumoto and J. Novak, <a href="http://arxiv.org/abs/0905.1992">Jucys-Murphy Elements and Unitary Matrix Integrals</a> arXiv.0905.1992 [math.CO], 2009-2012.
%F A135920 From _Peter Bala_, Sep 27 2012: (Start)
%F A135920 Let E(x) = cosh(sqrt(2*x)) = Sum_{n >= 0} x^n/((2*n)!/2^n). A generating function is E((E(x) - 1)) = 1 + x + 2*x^2/6 + 7*x^3/90 + ..., where the sequence of denominators [1, 1, 6, 90, ...] is given by (2*n)!/2^n. Cf. A000110 which has generating function exp((exp(x) - 1)).
%F A135920 An e.g.f. is E((E(x^2/2) - 1)) = 1 + x^2/2! + 2*x^4/4! + 7*x^6/6! + .... (End)
%F A135920 G.f.: 1 + x/(U(0)-x) where U(k) = 1 - 2*x*k - x*k^2 + x*(x*(k+1)^2 - 1)/U(k+1) ; (continued fraction). - _Sergei N. Gladkovskii_, Oct 11 2012
%F A135920 G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-k^2*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jan 16 2013
%F A135920 Conjecture: a(n) = A369527(n-1, 0) = A369595(n-1, 0) for n > 0 with a(0) = 1. - _Mikhail Kurkov_, Apr 25 2024
%e A135920 O.g.f.: A(x) = 1 + x/(1-x) + x^2/((1-x)*(1-4x)) + x^3/((1-x)*(1-4x)*(1-9x)) + x^4/((1-x)*(1-4x)*(1-9x)*(1-16x)) + ...
%e A135920 Also generated by iterated binomial transforms in the following way:
%e A135920 [1,2,7,37,264,2433,27913,...] = BINOMIAL([1,1,4,21,151,1422,16629,..]);
%e A135920 [1,4,21,151,1422,16629,234529,...] = BINOMIAL^3([1,1,6,43,393,4596,...]);
%e A135920 [1,6,43,393,4596,66049,1125905,...] = BINOMIAL^5([1,1,8,73,811,11274,...]);
%e A135920 [1,8,73,811,11274,191685,...] = BINOMIAL^7([1,1,10,111,1453,23328,...]);
%e A135920 [1,10,111,1453,23328,456033,...] = BINOMIAL^9([1,1,12,157,2367,43014,...]);
%e A135920 etc.
%t A135920 nmax = 20;
%t A135920 A[x_] = Sum[x^n/Product[1 - k^2 x, {k, 0, n}], {n, 0, nmax}];
%t A135920 CoefficientList[A[x] + O[x]^nmax, x] (* _Jean-François Alcover_, Jul 27 2018 *)
%o A135920 (PARI) a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-j^2*x+x*O(x^n))), n)
%Y A135920 Cf. A135921, A124373, A000110, A036969.
%K A135920 nonn
%O A135920 0,3
%A A135920 _Paul D. Hanna_, Dec 06 2007