OFFSET
0,2
COMMENTS
Row n has 1 + floor(n^2/3) terms.
Row sums are equal to 3^n = A000244(n).
Alternating row sums are 3^(ceiling(n/2)) = A108411(n+1).
T(n,0) = (n+1)*(n+2)/2 = A000217(n+1).
Sum_{k>=0} k*T(n,k) = 3^(n-1)*n*(n-1)/2 = A129530(n).
This sequence is mentioned in the Andrews-Savage-Wilf paper. - Omar E. Pol, Jan 30 2012
REFERENCES
M. Bona, Combinatorics of Permutations, Chapman & Hall/CRC, Boca Raton, FL, 2004, pp. 57-61.
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.
LINKS
Alois P. Heinz, Rows n = 0..50, flattened
G. E. Andrews, C. D. Savage and H. S. Wilf, Hypergeometric identities associated with statistics on words
Mark A. Shattuck and Carl G. Wagner, Parity Theorems for Statistics on Lattice Paths and Laguerre Configurations, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.1.
FORMULA
Generating polynomial of row n is Sum_{i=0..n} Sum_{j=0..n-i} binomial[n; i,j,n-i-j], where binomial[n;a,b,c] (a+b+c=n) is a q-multinomial coefficient.
Sum_{k=0..floor(n^2/3)} (-1)^k * T(n,k) = A056449(n). - Alois P. Heinz, Feb 12 2025
EXAMPLE
T(3,2) = 8 because we have 100, 110, 120, 200, 201, 211, 220 and 221.
Triangle starts:
1;
3;
6, 3;
10, 8, 8, 1;
15, 15, 21, 18, 9, 3;
21, 24, 39, 45, 48, 30, 24, 9, 3;
...
MAPLE
for n from 0 to 40 do br[n]:=sum(q^i, i=0..n-1) od: for n from 0 to 40 do f[n]:=simplify(product(br[j], j=1..n)) od: mbr:=(n, a, b, c)->simplify(f[n]/f[a]/f[b]/f[c]): for n from 0 to 9 do G[n]:=sort(simplify(sum(sum(mbr(n, a, b, n-a-b), b=0..n-a), a=0..n))) od: for n from 0 to 9 do seq(coeff(G[n], q, j), j=0..floor(n^2/3)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, l) option remember; `if`(n=0, 1, add(expand(b(n-1, `if`(j<3,
subsop(j=l[j]+1, l), l)))*x^([0, l[1], l[1]+l[2]][j]), j=1..3))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$2])):
seq(T(n), n=0..10); # Alois P. Heinz, Feb 12 2025
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Apr 22 2007
STATUS
approved