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A135920 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - k^2*x). 6
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'} is a subset of Vj.
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)
LINKS
S. Matsumoto and 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). - 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) )); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
From Mikhail Kurkov, Jan 14 2024: (Start)
a(n) = R_1(n-1, 0) for n > 0 with a(0) = 1 where R_1(n, q) = R_1(n-1, q+2) + Sum_{j=0..q} binomial(q, j)*R_1(n-1, j) for n > 0, q >= 0 with R_1(0, q) = 1 for q >= 0. If we change R_1(n-1, q+2) to R_1(n-1, q+1), we get Bell numbers.
a(n) = R_2(n, 0) where R_2(n, q) = q^2*R_2(n-1, q) + R_2(n-1, q+1) for n > 0, q >= 0 with R_2(0, q) = 1 for q >= 0. If we change q^2 to q, we get Bell numbers. (End)
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.
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)
(PARI) upto(n) = my(v1, v2, v3); v1 = vector(n + 1, i, 1); v2 = v1; v3 = vector(n + 1, i, 0); v3[1] = 1; for(i = 1, n, for(q = 0, n - i, v2[q + 1] = q ^ 2 * v1[q + 1] + v1[q + 2]); v1 = v2; v3[i + 1] = v1[1]); v3 \\ Mikhail Kurkov, Jan 14 2024
CROSSREFS
Sequence in context: A322140 A339459 A125515 * A001515 A144301 A036247
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 06 2007
STATUS
approved

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Last modified February 28 07:05 EST 2024. Contains 370387 sequences. (Running on oeis4.)