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

0,9

Links

G. C. Greubel, Rows n = 0..20 of the irregular triangle, flattened

Wikipedia, Major index

Formula

T(n,k) = A008302(n+1,k) for n + 1 <= k <= n*(n + 1)/2, n > 1. - Franck Maminirina Ramaharo, Oct 14 2018

Example

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;
    ...

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

Prog

(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 */
(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

Crossrefs

Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217.

Sequence in context: A238684 A352377 A202694 * A197032 A321781 A254862

Adjacent sequences: A123218 A123219 A123220 * A123222 A123223 A123224

Keyword

nonn,tabf

Author

Roger L. Bagula, Oct 05 2006

Extensions

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 14 2018

Status

approved