OFFSET
0,5
COMMENTS
Row n has 1 + floor(n/2) terms. Row sums are the partition numbers (A000041).
Sum_{k=0..floor(n/2)}k*T(n,k) = A066898(n).
From Gregory L. Simay, Nov 02 2015: (Start)
If n<=2k+1, T(n+2k,k) = A000041(n), the number of partitions of n.
T(n+2k,k) = the convolution of A000009(n-2j),which are the strict partitions of (n-2j), and p(j+k,k), which are the number of partitions of j+k having exactly k parts.
T(n+2k,k) = e(n,k) where e(n,0)= A000009(n) and e(n,k) = e(n,k-1) + e(n-2k,k-1) + e(n-4k,k-1) + ... .(End)
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
FORMULA
G.f.: G(t,x) = 1/Product_{j>=1}((1-x^(2j-1))(1-tx^(2j))).
From Gregory L. Simay, Nov 03 2015: (Start)
G.f.: T(n+2k,k) = g.f.: e(n,k) = Product_{j>=1}(1-x^2*(k+j))*p(x), where p(x) is the g.f. of the partitions of x. If n<=2k+1, then the g.f. reduces to p(x).
T(n+2k,k) = T(n+2k-2,k-1) + T(n,k).
(End)
EXAMPLE
T(7,2) = 3 because we have [4,2,1], [3,2,2] and [2,2,1,1,1].
Triangle starts:
1;
1;
1, 1;
2, 1;
2, 2, 1;
3, 3, 1;
4, 4, 2, 1;
5, 6, 3, 1;
6, 8, 5, 2, 1;
8, 11, 7, 3, 1;
10, 14, 10, 5, 2, 1;
12, 19, 14, 7, 3, 1;
15, 24, 19, 11, 5, 2, 1;
18, 31, 26, 15, 7, 3, 1;
22, 39, 34, 21, 11, 5, 2, 1;
27, 49, 45, 29, 15, 7, 3, 1;
Added entries for n=8 through n=15. - Gregory L. Simay, Nov 03 2015
From Gregory L. Simay, Nov 03 2015: (Start)
T(15,4) = T(7+2*4,4) = p(7) = 15, since 7 < 2*4 + 1.
T(15,3) = T(13,2) + T(9,3) = 26 + 3 = 29.
T(10,1) = T(8+2*1,1) = T(8,0) + T(6,0) + T(4,0) + T(2,0) + T(0,0) = 6 + 4 + 2 + 1 + 1 = 14.
T(15,3) = T(9+2*3) = e(9,3) = e(9,2) + e(3,2) = (e(9,1) + e(5,1) + e(1,1)) + e(3,1) = q(9) + q(7) + q(5) + q(3) + q(1) + q(5) + q(3) + q(1) + q(1) + q(3) + q(1) = q(9) + q(7) + 2*q(5) + 3*q(3) + 4*q(1) = 8 + 5 + 2*3 + 3*2 + 4*1 = 29 = the convolution sum of q(9-2j) with p(3+j,3).
(End)
MAPLE
g:=1/product((1-x^(2*j-1))*(1-t*x^(2*j)), j=1..20): gser:=simplify(series(g, x=0, 22)): P[0]:=1: for n from 1 to 18 do P[n]:=coeff(gser, x^n) od: for n from 0 to 18 do seq(coeff(P[n], t, j), j=0..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; local j; if n=0 then 1 elif i<1
then 0 else []; for j from 0 to n/i do zip((x, y)->x+y, %,
[`if`(irem(i, 2)=0, 0$j, [][]), b(n-i*j, i-1)], 0) od; %[] fi
end:
T:= n-> b(n, n):
seq (T(n), n=0..30); # Alois P. Heinz, Jan 07 2013
MATHEMATICA
nn=8; CoefficientList[Series[Product[1/(1-x^(2i-1))/(1-y x^(2i)), {i, 1, nn}], {x, 0, nn}], {x, y}]//Grid (* Geoffrey Critzer, Jan 07 2013 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Feb 17 2006
STATUS
approved