login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A081285 Table of coefficients of polynomials f_n(q) defined by 1/Product_{i>=1} (1-a q^i)^i = Sum_{n>=0} a^n q^n f_n(q) / ((q)_n)^2, where (x)_n is the q-Pochhammer symbol, defined to be Product_{i=0..n-1} (1-x q^i). 2
1, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 2, 4, 2, 4, 2, 4, 2, 1, 0, 1, 1, 0, 1, 2, 4, 6, 7, 8, 12, 12, 14, 12, 12, 8, 7, 6, 4, 2, 1, 0, 1, 1, 0, 1, 2, 4, 6, 12, 12, 21, 26, 37, 40, 55, 52, 61, 60, 61, 52, 55, 40, 37, 26, 21, 12, 12, 6, 4, 2, 1, 0, 1, 1, 0, 1, 2, 4, 6, 12, 18, 26, 38, 57, 76 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

f_n has degree n(n-1), so n-th row of table has n(n-1)+1 entries. Each row is palindromic. The sum of the terms in the n-th row is n!. The first n+1 terms of the n-th row are the same as the first n terms of A052847.

The 'major index' maj(p) of a permutation p = a_1 a_2 ... a_n is the sum of all i such that a_i > a_(i+1). f_n(q) = Sum_p q^(maj(p)+maj(p^(-1))), where the sum is over all permutations of {1,2,...,n}.

REFERENCES

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; Exercise 4.20.

LINKS

Alois P. Heinz, Rows n = 0..40, flattened

Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

FORMULA

f_n(q) = Sum_{r=1..n} (-1)^(r+1) q^(r(r-1)/2) (q)_(n-1) (q)_n / ((q)_(r) ((q)_(n-r))^2) f_(n-r)(q) for n>=1.

EXAMPLE

f_0 = f_1 = 1, f_2 = 1+q^2, f_3 = 1+q^2+2q^3+q^4+q^6, so sequence begins 1; 1; 1,0,1; 1,0,1,2,1,0,1; ...

MAPLE

b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,

      add(b(u-j, o+j-1)*x^(-u), j=1..u)+

      add(b(u+j-1, o-j)*x^( o), j=1..o)))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n, 0)):

seq(T(n), n=0..10);  # Alois P. Heinz, Apr 28 2018

MATHEMATICA

qpoch[x_, n_] := Product[1-x*q^i, {i, 0, n-1}]; f[0]=1; f[n_] := f[n]=Together[Sum[ -(-1)^r q^Binomial[r, 2] qpoch[q^(n-r+1), r-1]*qpoch[q^(r+1), n-r]/qpoch[q, n-r] f[n-r], {r, 1, n}]]; Join@@Table[CoefficientList[f[n], q], {n, 0, 7}]

CROSSREFS

Row sums give A000142.

Cf. A052847.

Sequence in context: A044933 A295343 A025915 * A255361 A069844 A233006

Adjacent sequences:  A081282 A081283 A081284 * A081286 A081287 A081288

KEYWORD

nonn,tabf,easy

AUTHOR

Dean Hickerson, using information supplied by Moshe Shmuel Newman and Richard Stanley, Mar 15 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 20 22:22 EDT 2019. Contains 322310 sequences. (Running on oeis4.)