OFFSET
0,6
COMMENTS
T(n^2,n) = T(n^2+n,-n) = n! = A000142(n) for n>=0.
LINKS
Alois P. Heinz, Rows n = 0..500, flattened
EXAMPLE
T(8,-1) = 15: [2,2,2,2], [1,1,2,4], [1,1,4,2], [1,2,1,4], [1,2,4,1], [1,4,1,2], [1,4,2,1], [2,1,1,4], [2,1,4,1], [2,4,1,1], [4,1,1,2], [4,1,2,1], [4,2,1,1], [4,4], [8].
Triangle T(n,k) begins:
: n\k : -3 -2 -1 0 1 2 3 ...
+-----+------------------------------------
: 0 : 1;
: 1 : 1;
: 2 : 1, 0, 1;
: 3 : 2, 2;
: 4 : 2, 3, 1, 2;
: 5 : 11, 2, 3;
: 6 : 2, 2, 14, 8, 6;
: 7 : 6, 33, 14, 11;
: 8 : 5, 15, 43, 45, 20;
: 9 : 44, 82, 99, 25, 6;
: 10 : 14, 74, 141, 230, 41, 12;
: 11 : 202, 260, 451, 85, 26;
: 12 : 6, 22, 351, 514, 953, 148, 54;
: 13 : 24, 766, 1049, 1798, 355, 104;
: 14 : 18, 104, 1301, 2321, 3503, 751, 194;
MAPLE
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
expand(add(`if`(j=0, 1, x^(2*irem(i, 2)-1))*
b(n-i*j, i-1, p+j)/j!, j=0..n/i))))
end:
T:= n->(p->seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..20);
MATHEMATICA
b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, Expand[Sum[If[j==0, 1, x^(2*Mod[i, 2]-1)]*b[n-i*j, i-1, p+j]/j!, {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, Exponent[p, x, Min], Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jan 17 2017, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, May 19 2014
STATUS
approved