

A016753


Expansion of 1/((13*x)*(14*x)*(15*x)).


3



1, 12, 97, 660, 4081, 23772, 133057, 724260, 3863761, 20308332, 105558817, 544039860, 2785713841, 14192221692, 72020501377, 364354427460, 1838822866321, 9262446387852, 46585947584737
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OFFSET

0,2


COMMENTS

As (0,0,1,12,97,...) this is the fourth binomial transform of cosh(x)1. It is the binomial transform of A016269, when this has two leading zeros. Its e.g.f. is then exp(4x)cosh(x)  exp(4x).  Paul Barry, May 13 2003
This gives the third column of the Sheffer triangle A143495 (3restricted Stirling2 numbers). See the e.g.f. below, and A193685 for comments on the general case.  Wolfdieter Lang, Oct 08 2011
From Kevin Long, Mar 25 2017: (Start)
In the power set poset 2^(n+2), a(n) gives the number of size 3 subposets {A,B,C} such that A subset of C, B subset of C, and AB. By symmetry, it also counts the size 3 subposets {A,B,C} such that C subset of A, C subset of B, and AB.
By the power set poset, I mean the subsets of [n+2] ordered by inclusion. AB means A and B are incomparable.
The result can be proved by showing that the formula holds. 5^n counts triples (A,B,C) of subsets of [n] where A subset of C and B subset of C, since for each x in [n], it is either in C only, in A and C, in B and C, in all three, or in none. However, this also counts the cases where A subset of B and where B subset of A, and we want AB.
Each case can be counted by 4^n, since if A subset of B⊆C, then each element x of [n] is either in all three, in B and C, in only C, or in none. Hence we subtract 2*4^n from 5^n. These two cases intersect, however, when A = B subset of C, which can be counted by 3^n, since each element x of [n] can be either in all three sets, in only C, or in none.
For the purposes of inclusionexclusion, we add these sets back in to get 5^n2*4^n+3^n to count all triples (A,B,C) where A subset of C, B subset of C, and AB. We want sets, not triples, so this doublecounts the sets since interchanging A and B give the same set, so we divide this by 2. Hence the formula for a(n) counts these subposets for 2^(n+2). (End)


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,47,60).


FORMULA

a(n) = 5^(n+2)/2  4^(n+2) + 3^(n+2)/2.  Paul Barry, May 13 2003
If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*stirling2(k,j)*x^(mk) then a(n2) = f(n,2,3), (n >= 2).  Milan Janjic, Apr 26 2009
a(n) = 9*a(n1)  20*a(n2) + 3^n, n >= 2.  Vincenzo Librandi, Mar 20 2011
O.g.f.: 1/((13*x)*(14*x)*(15*x)).
E.g.f.: (d^2/dx^2) (exp(3*x)*((exp(x)1)^2)/2!).  Wolfdieter Lang, Oct 08 2011
a(n) = A245019(n+2)/2.  Kevin Long, Mar 24 2017


MATHEMATICA

CoefficientList[ Series[ 1/((1  3x)(1  4x)(1  5x)), {x, 0, 25} ], x ]
LinearRecurrence[{12, 47, 60}, {1, 12, 97}, 30] (* G. C. Greubel, Sep 15 2018 *)


PROG

(PARI) Vec(1/((13*x)*(14*x)*(15*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012
(MAGMA) [(5^(n+2)  2*4^(n+2) + 3^(n+2))/2: n in [0..30]]; // G. C. Greubel, Sep 15 2018


CROSSREFS

Cf. A000244, A005061, A016753, A132495, A193685, A245019.
Sequence in context: A251430 A027255 A121791 * A078605 A021029 A270496
Adjacent sequences: A016750 A016751 A016752 * A016754 A016755 A016756


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



