A130523 Square array, read by antidiagonals, where row n+1 equals the partial sums of the previous row after removing the n-th term from row n for n>=0, with row 0 equal to all 1's.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 5, 8, 4, 1, 1, 6, 18, 13, 5, 1, 1, 7, 24, 37, 19, 6, 1, 1, 8, 31, 87, 63, 26, 7, 1, 1, 9, 39, 118, 184, 97, 34, 8, 1, 1, 10, 48, 157, 442, 324, 140, 43, 9, 1, 1, 11, 58, 205, 599, 959, 517, 193, 53, 10, 1, 1, 12, 69, 263, 804, 2332, 1733, 774, 257, 64, 11

Offset: 0

Author

Paul D. Hanna, Jun 02 2007, Jun 06 2007

Keywords

• nonn
• tabl

Comments

The g.f. of n-th lower diagonal equals D(x)*F(x)*C(x)^n and the g.f. of n-th upper diagonal equals D(x)*F(x)^n, where D(x) is g.f. of main diagonal (A007857), C(x) is g.f. of Catalan numbers (A000108) and F(x) is g.f. of ternary numbers (A001764).

Examples

			Square array begins:
  (1), 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
  1, (2), 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...;
  1, 4, (8), 13, 19, 26, 34, 43, 53, 64, 76, 89, 103, 118, 134, ...;
  1, 5, 18, (37), 63, 97, 140, 193, 257, 333, 422, 525, 643, 777, ...;
  1, 6, 24, 87, (184), 324, 517, 774, 1107, 1529, 2054, 2697, 3474, ...;
  1, 7, 31, 118, 442, (959), 1733, 2840, 4369, 6423, 9120, 12594, ...;
  1, 8, 39, 157, 599, 2332, (5172), 9541, 15964, 25084, 37678, ...;
  1, 9, 48, 205, 804, 3136, 12677, (28641), 53725, 91403, 146077, ...;
  1, 10, 58, 263, 1067, 4203, 16880, 70605, (162008), 308085, ...;
  1, 11, 69, 332, 1399, 5602, 22482, 93087, 401172, (932503), ...;
  ...
For each row, remove the term along the diagonal (in parenthesis here),
and then take partial sums to obtain the next row.
		

Crossrefs

Cf. Diagonals: A007857, A130524, A130525; related: A000108, A001764.

Programs

PARI

{T(n,k) = if(n<0||k<0,0,if(n==0,1,if(n>k+1, T(n,k-1) + T(n-1,k), T(n,k-1) + T(n-1,k+1))))}
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))

PARI

/* Using Formula for G.F.: */
{T(n,k) = local(m=max(n,k)+1,C,F,D); C=subst(Ser(vector(m,r,binomial(2*r-2,r-1)/r)),x,x*y); F=subst(Ser(vector(m,r,binomial(3*r-3,r-1)/(2*r-1))),x,x*y); D=1/(1-x*y*C*F-x*y*F^2);A=D*(1/(1-y*F) + x*C*F/(1-x*C)); polcoeff(polcoeff(A+O(x^m),n,x)+O(y^m),k,y)}
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))

Formula

G.f.: A(x,y) = D(x*y)*( 1/(1 - y*F(x*y)) + x*C(x*y)*F(x*y)/(1 - x*C(x*y)) ), where D(x) = 1/(1 - x*C(x)*F(x) - x*F(x)^2) is the g.f. of the main diagonal (A007857), C(x) = 1 + x*C(x)^2 is the g.f. of Catalan numbers (A000108) and F(x) = 1 + x*F(x)^3 is the g.f. of ternary numbers (A001764).