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

0,5

Links

Table of n, a(n) for n=0..72.

Formula

G.f.: A(x,y) = D(xy)/[(1 - yF(xy)) + xC(xy)F(xy)^2/(1 - xC(xy))], where D(x) = 1/[1 - xC(x)F(x)^2 - xF(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.

Example

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 + xC(x)^2 is g.f. of Catalan numbers (A000108):
[1, 1, 2, 5, 14, 42, 132, 429, 1430, ..., C(2n,n)/(n+1), ...] and
F(x) = 1 + xF(x)^4 is g.f. of A002293:
[1, 1, 4, 22, 140, 969, 7084, 53820, ..., C(4n,n)/(3n+1), ...].

Prog

(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)

Crossrefs

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

Sequence in context: A100398 A160364 A107735 * A079213 A047884 A124328

Adjacent sequences: A137567 A137568 A137569 * A137571 A137572 A137573

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 27 2008

Status

approved