A110582 Triangle, read by rows, where the g.f. of diagonal n, D_n(x), is generated from the g.f. of row n-1, R_{n-1}(x), by D_n(x) = R_{n-1}(x)/(1-x)^2 for n>0, with D_0(x) = 1/(1-x)^2.

1, 1, 2, 1, 2, 3, 1, 4, 3, 4, 1, 4, 7, 4, 5, 1, 6, 10, 10, 5, 6, 1, 6, 14, 16, 13, 6, 7, 1, 8, 18, 26, 22, 16, 7, 8, 1, 8, 25, 34, 38, 28, 19, 8, 9, 1, 10, 29, 52, 55, 50, 34, 22, 9, 10, 1, 10, 37, 66, 84, 76, 62, 40, 25, 10, 11, 1, 12, 44, 90, 116, 122, 97, 74, 46, 28, 11, 12

Offset: 0

Author

Paul D. Hanna, Jul 29 2005

Keywords

• nonn
• tabl

Comments

Related to planar partitions.

Examples

			Triangle begins:
1;
1,2;
1,2,3;
1,4,3,4;
1,4,7,4,5;
1,6,10,10,5,6;
1,6,14,16,13,6,7;
1,8,18,26,22,16,7,8;
1,8,25,34,38,28,19,8,9;
1,10,29,52,55,50,34,22,9,10; ...
Row sums form A006330 (offset 1):
{1,3,6,12,21,38,63,106,170,272,422,653,...},
(planar partitions with only one row and one column).
G.f. of diagonal n, D_n(x), is generated from g.f. of
row n-1, R_{n-1}(x), by D_n(x) = R_{n-1}(x)/(1-x)^2:
D_3(x) = 1 + 4*x + 10*x^2 + 16*x^3 + 22*x^4 + ...
= (1 + 2*x + 3*x^2)/(1-x)^2 = R_2(x)/(1-x)^2;
D_4(x) = 1 + 6*x + 14*x^2 + 26*x^3 + 38*x^4 + ...
= (1+ 4*x+ 3*x^2+ 4*x^3)/(1-x)^2 = R_3(x)/(1-x)^2.
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A006330 (row sums).

Programs

Mathematica

T[n_, k_] := T[n, k] = If[n < k || k < 0, 0, If[k == 0, 1, If[k == n, n + 1, Sum[T[n - k - 1, j]*(k - j + 1), {j, 0, k}]]]]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Sep 01 2017 *)

PARI

T(n,k)=if(n
				

PARI

T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k));if(n
				

Formula

T(n, k) = Sum_{j=0..k} T(n-k-1, j)*(k-j+1) with T(n, n) = n+1.

G.f.: A(x, y) = Sum_{j=0..n} x^j/Product_{i=1..j+1} (1-y*x^i)^2.