OFFSET
1,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..250
F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger, The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r, arXiv:1504.02513, 2015.
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.
Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
FORMULA
a(n) ~ 9 * 5^(2*n + 25/2) / (512 * n^12 * Pi^2). - Vaclav Kotesovec, Sep 10 2014
MAPLE
h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
+add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
end:
g:= proc(n, i, l)
`if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
end:
a:= n-> g(n, 5, []):
seq(a(n), n=1..30); # Alois P. Heinz, Apr 10 2012
# second Maple program
a:= proc(n) option remember; `if`(n<6, n!, ((-375+400*n+843*n^2
+322*n^3+35*n^4)*a(n-1) +225*(n-1)^2*(n-2)^2*a(n-3)
-(259*n^2+622*n+45)*(n-1)^2*a(n-2))/ ((n+6)^2*(n+4)^2))
end:
seq(a(n), n=1..30); # Alois P. Heinz, Sep 26 2012
MATHEMATICA
h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1+l[[i]]-j+Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]]^2, If[i<1, 0, Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_, k_] := If[k >= n, n!, g[n, k, {}]]; Table[a[n, 5], {n, 1, 30}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric Rains (rains(AT)caltech.edu), N. J. A. Sloane
EXTENSIONS
More terms from Naohiro Nomoto, Mar 01 2002
More terms from Alois P. Heinz, Apr 10 2012
STATUS
approved