OFFSET
0,8
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux
Wikipedia, Young tableau
FORMULA
G.f. of column k: 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)).
A(n,k) = Sum_{i=0..k} C(k,i) * A138177(n,k-i). - Alois P. Heinz, Apr 06 2015
EXAMPLE
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 1, 4, 9, 16, 25, 36, ...
0, 1, 6, 19, 44, 85, 146, ...
0, 1, 9, 39, 116, 275, 561, ...
0, 1, 12, 69, 260, 751, 1812, ...
0, 1, 16, 119, 560, 1955, 5552, ...
MAPLE
# First program:
h:= (l, k)-> mul(mul((k+j-i)/(1+l[i] -j +add(`if`(l[t]>=j, 1, 0)
, t=i+1..nops(l))), j=1..l[i]), i=1..nops(l)):
g:= proc(n, i, k, l)
`if`(n=0, h(l, k), `if`(i<1, 0, g(n, i-1, k, l)+
`if`(i>n, 0, g(n-i, i, k, [l[], i]))))
end:
A:= (n, k)-> `if`(n=0, 1, g(n, n, k, [])):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second program:
gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)):
A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
MATHEMATICA
(* First program: *)
h[l_, k_] := Product[Product[(k+j-i)/(1+l[[i]]-j + Sum[If[l[[t]] >= j, 1, 0], {t, i+1, Length[l]}]), {j, 1, l[[i]]}], {i, 1, Length[l]}]; g [n_, i_, k_, l_] := If[n == 0, h[l, k], If[i < 1, 0, g[n, i-1, k, l] + If[i > n, 0, g[n-i, i, k, Append[l, i]]]]]; a[n_, k_] := If[n == 0, 1, g[n, n, k, {}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten
(* second program: *)
gf[k_] := 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)); a[n_, k_] := Coefficient[Series[gf[k], {x, 0, n+1}], x, n]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Mar 20 2012
STATUS
approved