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

0,5

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).

Links

Table of n, a(n) for n=0..76.

Formula

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

Example

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.

Prog

(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(""))

Crossrefs

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

Sequence in context: A320796 A026725 A026758 * A034363 A368735 A026769

Adjacent sequences: A130520 A130521 A130522 * A130524 A130525 A130526

Keyword

nonn,tabl

Author

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

Status

approved