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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1325; terms in rows 0..50.

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).

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).

Example

Square array begins:
  (1), 1,  1,   1,    1,    1,     1,      1,       1, ...;
  1,  (2), 3,   4,    5,    6,     7,      8,       9, ...;
  1,   4, (8), 13,   19,   26,    34,     43,      53, ...;
  1,   5, 18, (37),  63,   97,   140,    193,     257, ...;
  1,   6, 24,  87, (184), 324,   517,    774,    1107, .,.;
  1,   7, 31, 118,  442, (959), 1733,   2840,    4369, ...;
  1,   8, 39, 157,  599, 2332, (5172),  9541,   15964, ...;
  1,   9, 48, 205,  804, 3136, 12677, (28641),  53725, ...;
  1,  10, 58, 263, 1067, 4203, 16880,  70605, (162008), ...;
  ...
For each row, remove the term along the main diagonal (A007857, shown in parenthesis), and then take partial sums to obtain the next row.
G.f.: A(x,y) = 1 + (1*x + 1*y) + (1*x^2 + 2*x*y + 1*y^2) + (1*x^3 + 4*x^2*y + 3*x*y^2 + 1*y^3) + (1*x^4 + 5*x^3*y + 8*x^2*y^2 + 4*x*y^3 + 1*y^4) + (1*x^5 + 6*x^4*y + 18*x^3*y^2 + 13*x^2*y^3 + 5*x*y^4 + 1*y^5) + (1*x^6 + 7*x^5*y + 24*x^4*y^2 + 37*x^3*y^3 + 19*x^2*y^4 + 6*x*y^5 + 1*y^6) + ...
which may also be written as
A(x,y) = 1/(1-y) + x/(1-y)^2 + x^2*(1 + y - y^2)/(1-y)^3 + x^3*(1 + y + 4*y^2 - 9*y^3 + 4*y^4)/(1-y)^4 + x^4*(1 + y + 4*y^2 + 17*y^3 - 66*y^4 + 63*y^5 - 19*y^6)/(1-y)^5 + x^5*(1 + y + 4*y^2 + 17*y^3 + 74*y^4 - 444*y^5 + 673*y^6 - 422*y^7 + 97*y^8)/(1-y)^6 + x^6*(1 + y + 4*y^2 + 17*y^3 + 74*y^4 + 330*y^5 - 2864*y^6 + 6075*y^7 - 5898*y^8 + 2778*y^9 - 517*y^10)/(1-y)^7 + ...

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) = my(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)); polcoef(polcoef(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 A378809 A368735

Adjacent sequences: A130520 A130521 A130522 * A130524 A130525 A130526

Keyword

nonn,tabl

Author

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

Status

approved