A137570 Square array, read by antidiagonals, where row n+1 equals the partial sums of the previous row after removing the terms in positions {n, n+1} from row n for n>=0, with row 0 equal to all 1's.

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 6, 10, 4, 1, 1, 7, 29, 16, 5, 1, 1, 8, 36, 60, 23, 6, 1, 1, 9, 44, 186, 100, 31, 7, 1, 1, 10, 53, 230, 397, 150, 40, 8, 1, 1, 11, 63, 283, 1281, 681, 211, 50, 9, 1, 1, 12, 74, 346, 1564, 2802, 1051, 284, 61, 10, 1, 1, 13, 86, 420, 1910, 9294, 4908

Offset: 0

Author

Paul D. Hanna, Jan 27 2008

Keywords

• nonn
• tabl

Examples

			Square array begins:
  (1),(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, 5,(10),(16), 23, 31, 40, 50, 61, 73, 86, 100, 115, 131, 148, ...;
  1, 6, 29,(60),(100), 150, 211, 284, 370, 470, 585, 716, 864, ...;
  1, 7, 36, 186,(397),(681), 1051, 1521, 2106, 2822, 3686, 4716, ...;
  1, 8, 44, 230, 1281,(2802),(4908), 7730, 11416, 16132, 22063, ...;
  1, 9, 53, 283, 1564, 9294,(20710),(36842), 58905, 88319, 126730, ...;
  1, 10, 63, 346, 1910, 11204, 70109,(158428),(285158), 461190, ...;
  1, 11, 74, 420, 2330, 13534, 83643, 544833,(1244413),(2260257), ...;
  ...
For each row, remove the terms along the diagonals (in parenthesis),
and then take partial sums to obtain the next row.
GENERATING FUNCTIONS.
The g.f. of n-th lower diagonal equals D(x)*F(x)^2*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 (A137571):
[1, 2, 10, 60, 397, 2802, 20710, 158428, 1244413, 9980220, ...]
defined by:
D(x) = 1/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where
C(x) = 1 + x*C(x)^2 is g.f. of Catalan numbers (A000108):
[1, 1, 2, 5, 14, 42, 132, 429, 1430, ..., C(2*n,n)/(n+1), ...] and
F(x) = 1 + x*F(x)^4 is g.f. of A002293:
[1, 1, 4, 22, 140, 969, 7084, 53820, ..., C(4*n,n)/(3*n+1), ...].
		

Crossrefs

Cf. A130523 (variant); diagonals: A137571, A137572, A137573; related: A000108, A002293.

Programs

PARI

T(n, k)=if(k<0, 0, if(n==0, 1, T(n, k-1) + if(n-1>k, T(n-1, k), T(n-1, k+2))))

PARI

/* Using Formula for G.F.: */ T(n,k)=local(m=max(n,k)+1,C,F,D,A); C=subst(Ser(vector(m,r,binomial(2*r-2,r-1)/r)),x,x*y); F=subst(Ser(vector(m,r,binomial(4*r-4,r-1)/(3*r-2))),x,x*y); D=1/(1-x*y*C*F^2-x*y*F^3); A=D*(1/(1-y*F) + x*C*F^2/(1-x*C)); polcoeff(polcoeff(A+O(x^m),n,x)+O(y^m),k,y)

Formula

G.f.: A(x,y) = D(x*y)*(1/(1 - y*F(x*y)) + x*C(x*y)*F(x*y)^2/(1 - x*C(x*y))), where D(x) = 1/(1 - x*C(x)*F(x)^2 - x*F(x)^3) is the g.f. of the main diagonal (A137571), C(x) = g.f. of Catalan numbers (A000108) and F(x) = g.f. of A002293; thus the g.f. of n-th lower diagonal = D(x)*F(x)^2*C(x)^n and the g.f. of n-th upper diagonal = D(x)*F(x)^n.