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Search: id:A081285
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| A081285 |
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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). |
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| 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; listen)
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OFFSET
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0,9
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COMMENT
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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}.
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; Exercise 4.20.
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LINKS
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Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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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.
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EXAMPLE
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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; ...
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A052847.
Adjacent sequences: A081282 A081283 A081284 this_sequence A081286 A081287 A081288
Sequence in context: A070107 A044933 A025915 this_sequence A069844 A124327 A082596
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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Dean Hickerson (dean(AT)math.ucdavis.edu), using information supplied by Moshe Newman and R. P. Stanley, Mar 15 2003.
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