A171502 - OEIS

%I #13 Jul 19 2021 02:32:58

%S 1,1,1,1,1,1,2,3,4,4,4,3,2,1,1,3,6,10,14,18,21,23,23,21,18,14,10,6,3,

%T 1,1,4,10,20,34,52,73,96,119,140,157,168,172,168,157,140,119,96,73,52,

%U 34,20,10,4,1

%N Irregular triangle T(n, k) = coefficients of p(x, n), where p(x, n) = ((1-x^(2*n))/(1-x))*p(x, n-1), p(x, 0) = 1, and p(x, 1) = (1-x^4)/(1-x), read by rows.

%D Samuel I. Goldberg, Curvature and Homology, Dover, New York, 1998, page 144.

%H G. C. Greubel, <a href="/A171502/b171502.txt">Rows n = 1..20 of the irregular triangle, flattened</a>

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000059">The inversion number of a standard Young tableau as defined by Haglund and Stevens.</a>

%F T(n, k) = coefficients of p(x, n), where p(x, n) = ((1-x^(2*n))/(1-x))*p(x, n-1), p(x, 0) = 1, and p(x, 1) = (1-x^4)/(1-x).

%F T(n, k) = coefficients of f(n, x), where f(n, x) = (1/(1-x)^(n-1))*Product_{j=2..n} (1 - x^(2*j)). - _G. C. Greubel_, Jul 18 2021

%e Irregular triangle begins as:

%e   1;

%e   1, 1, 1,  1;

%e   1, 2, 3,  4,  4,  4,  3,  2,  1;

%e   1, 3, 6, 10, 14, 18, 21, 23, 23, 21, 18, 14, 10, 6, 3, 1;

%t p[n_, x_]:= p[n, x]= Product[1 - x^(2*j), {j, 2, n}]/(1-x)^(n-1);

%t T[n_]:= T[n] = CoefficientList[p[n, x], x];

%t Table[T[n], {n, 10}]//Flatten

%o (Sage)

%o @CachedFunction

%o def p(n,x): return product( 1-x^(2*j) for j in (2..n) )/(1-x)^(n-1)

%o def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)

%o flatten([T(n) for n in (1..12)]) # _G. C. Greubel_, Jul 18 2021

%Y Cf. A142724.

%K nonn,tabf

%O 1,7

%A _Roger L. Bagula_, Dec 10 2009

%E Edited by _G. C. Greubel_, Jul 18 2021