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Triangle, read by rows, where T(n,k) equals the number of permutations of {1..n+1} with [(nk+k)/2] inversions for n>=k>=0.
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%I #11 Dec 04 2023 10:05:07

%S 1,1,1,1,2,1,1,5,5,1,1,9,22,15,1,1,29,90,90,29,1,1,49,359,573,359,98,

%T 1,1,174,1415,3450,3450,1415,174,1,1,285,5545,17957,29228,21450,5545,

%U 628,1,1,1068,21670,110010,230131,230131,110010,21670,1068,1

%N Triangle, read by rows, where T(n,k) equals the number of permutations of {1..n+1} with [(nk+k)/2] inversions for n>=k>=0.

%C Row sums equal 2*n! for n>0.

%H Alois P. Heinz, <a href="/A128564/b128564.txt">Rows n = 0..140, flattened</a>

%F T(n,k) = A008302(n+1, [(nk+k)/2]) = coefficient of q^[(nk+k)/2] in the q-factorial of n+1 for n>=0.

%e Row sums equal 2*n! for n>0:

%e [1, 2, 4, 12, 48, 240, 1440, 10080, 80640, ..., 2*n!,...].

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 5, 5, 1;

%e 1, 9, 22, 15, 1;

%e 1, 29, 90, 90, 29, 1;

%e 1, 49, 359, 573, 359, 98, 1;

%e 1, 174, 1415, 3450, 3450, 1415, 174, 1;

%e 1, 285, 5545, 17957, 29228, 21450, 5545, 628, 1;

%e 1, 1068, 21670, 110010, 230131, 230131, 110010, 21670, 1068, 1;

%e 1, 1717, 84591, 526724, 1729808, 2409581, 1729808, 686763, 84591, 4015, 1;

%e ...

%p b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,

%p add(b(u+j-1, o-j)*x^(u+j-1), j=1..o)+

%p add(b(u-j, o+j-1)*x^(u-j), j=1..u)))

%p end:

%p T:= (n, k)-> coeff(b(n+1, 0), x, iquo((n+1)*k, 2)):

%p seq(seq(T(n,k), k=0..n), n=0..10); # _Alois P. Heinz_, May 02 2017

%t b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1, Sum[b[u + j - 1, o - j]* x^(u+j-1), {j, 1, o}] + Sum[b[u-j, o+j-1]*x^(u-j), {j, 1, u}]]];

%t T[n_, k_] := Coefficient[b[n+1, 0], x, Quotient[(n+1)*k, 2]];

%t Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Dec 06 2019, after _Alois P. Heinz_ *)

%o (PARI) {T(n,k)=local(faq=prod(j=1, n+1, (1-q^j)/(1-q))); polcoeff(faq, (n*k+k)\2, q)}

%Y Cf. A008302 (Mahonian numbers); A128565 (column 1), A128566 (column 2).

%Y Row sums give A098558.

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Mar 12 2007