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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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 A026769 A257365

Adjacent sequences:  A130520 A130521 A130522 * A130524 A130525 A130526

KEYWORD

nonn,tabl

AUTHOR

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

STATUS

approved

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Last modified April 25 16:59 EDT 2019. Contains 322461 sequences. (Running on oeis4.)