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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.
13
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
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
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