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A135920 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k^2*x). 4
1, 1, 2, 7, 37, 264, 2433, 27913, 386906, 6346119, 121159373, 2655174768, 66028903633, 1845579100993, 57506847262162, 1983312152411351, 75238783332550789, 3122408658986242072, 141063757638078429489 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

From Peter Bala, Sep 27 2012: (Start)

Generalized Bell numbers; row sums of A036969. 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′} ⊆ Vj. An example is given below.

(End)

LINKS

Table of n, a(n) for n=0..18.

S. Matsumoto, J. Novak, Jucys-Murphy Elements and Unitary Matrix Integrals arXiv.0905.1992 [math.CO], 2009-2012.

FORMULA

From Peter Bala, Sep 27 2012: (Start)

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)).

An e.g.f. is E((E(x^2/2)-1)) = 1 + x^2/2! + 2*x^4/4! + 7*x^6/6! + ....

(End)

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, 1-step). - Sergei N. Gladkovskii, Oct 11 2012

G.f.: (G(0) - 1)/(x-1) where G(k) =  1 - 1/(1-k^2*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013

EXAMPLE

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)) + ...

Also generated by iterated binomial transforms in the following way:

[1,2,7,37,264,2433,27913,...] = BINOMIAL([1,1,4,21,151,1422,16629,..]);

[1,4,21,151,1422,16629,234529,...] = BINOMIAL^3([1,1,6,43,393,4596,...]);

[1,6,43,393,4596,66049,1125905,...] = BINOMIAL^5([1,1,8,73,811,11274,...]);

[1,8,73,811,11274,191685,...] = BINOMIAL^7([1,1,10,111,1453,23328,...]);

[1,10,111,1453,23328,456033,...] = BINOMIAL^9([1,1,12,157,2367,43014,...]);

etc.

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'}.

MATHEMATICA

nmax = 20;

A[x_] = Sum[x^n/Product[1 - k^2 x, {k, 0, n}], {n, 0, nmax}];

CoefficientList[A[x] + O[x]^nmax, x] (* Jean-François Alcover, Jul 27 2018 *)

PROG

(PARI) a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-j^2*x+x*O(x^n))), n)

CROSSREFS

Cf. A135921, A124373. A000110, A036969.

Sequence in context: A072597 A322140 A125515 * A001515 A144301 A036247

Adjacent sequences:  A135917 A135918 A135919 * A135921 A135922 A135923

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 06 2007

STATUS

approved

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Last modified September 20 10:24 EDT 2020. Contains 337264 sequences. (Running on oeis4.)