A123221 Irregular triangle read by rows: the n-th row consists of the coefficients in the expansion of Sum_{j=0..n*(n+1)/2} A008302(n+1,j)*x^j*(1 - x)^(n - min(n, j)), where A008302 is the triangle of Mahonian numbers.
1, 1, 1, 0, 1, 1, 1, 0, 2, 3, 5, 3, 1, 1, 0, 3, 5, 11, 22, 20, 15, 9, 4, 1, 1, 0, 4, 7, 18, 41, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1, 1, 0, 5, 9, 26, 64, 154, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1, 1, 0, 6, 11, 35, 91, 234, 583
Offset: 0
Examples
Triangle begins:
1;
1;
1, 0, 1, 1;
1, 0, 2, 3, 5, 3, 1;
1, 0, 3, 5, 11, 22, 20, 15, 9, 4, 1;
1, 0, 4, 7, 18, 41, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1;
...
Links
- G. C. Greubel, Rows n = 0..20 of the irregular triangle, flattened
- Wikipedia, Major index
Programs
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Mathematica
M[n_]:= CoefficientList[Product[1-x^j, {j, n}]/(1-x)^n, x]; Table[CoefficientList[Sum[M[n+1][[m+1]]*x^m*(1-x)^(n -Min[n, m]), {m, 0, Binomial[n+1,2]}], x], {n, 0, 10}]//Flatten -
Maxima
A008302(n, k) := ratcoef(ratsimp(product((1 - x^j)/(1 - x), j, 1, n)), x, k)$ P(x, n) := sum(A008302(n + 1, j)*x^j*(1 - x)^(n - min(n, j)), j, 0, n*(n + 1)/2)$ create_list(ratcoef(expand(P(x, n)), x, k), n, 0, 10, k, 0, hipow(P(x, n), x)); /* Franck Maminirina Ramaharo, Oct 14 2018 */
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Sage
@CachedFunction def A008302(n,k): if (k<0 or k>binomial(n,2)): return 0 elif (n==1 and k==0): return 1 else: return A008302(n, k-1) + A008302(n-1, k) - A008302(n-1, k-n) def p(n,x): return sum( A008302(n+1, j)*x^j*(1-x)^(n-min(n, j)) for j in (0..binomial(n+1,2)) ) def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False) flatten([T(n) for n in (0..12)]) # G. C. Greubel, Jul 16 2021
Formula
T(n,k) = A008302(n+1,k) for n + 1 <= k <= n*(n + 1)/2, n > 1. - Franck Maminirina Ramaharo, Oct 14 2018
Extensions
Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 14 2018