A130523 - OEIS

%I #11 Jan 27 2025 06:48:24

%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,

%T 87,63,26,7,1,1,9,39,118,184,97,34,8,1,1,10,48,157,442,324,140,43,9,1,

%U 1,11,58,205,599,959,517,193,53,10,1,1,12,69,263,804,2332,1733,774,257,64,11

%N 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.

%C 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).

%F 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).

%e Square array begins:

%e   (1), 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;

%e   1, (2), 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...;

%e   1, 4, (8), 13, 19, 26, 34, 43, 53, 64, 76, 89, 103, 118, 134, ...;

%e   1, 5, 18, (37), 63, 97, 140, 193, 257, 333, 422, 525, 643, 777, ...;

%e   1, 6, 24, 87, (184), 324, 517, 774, 1107, 1529, 2054, 2697, 3474, ...;

%e   1, 7, 31, 118, 442, (959), 1733, 2840, 4369, 6423, 9120, 12594, ...;

%e   1, 8, 39, 157, 599, 2332, (5172), 9541, 15964, 25084, 37678, ...;

%e   1, 9, 48, 205, 804, 3136, 12677, (28641), 53725, 91403, 146077, ...;

%e   1, 10, 58, 263, 1067, 4203, 16880, 70605, (162008), 308085, ...;

%e   1, 11, 69, 332, 1399, 5602, 22482, 93087, 401172, (932503), ...;

%e   ...

%e For each row, remove the term along the diagonal (in parenthesis here),

%e and then take partial sums to obtain the next row.

%o (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))))}

%o for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))

%o (PARI) /* Using Formula for G.F.: */

%o {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)}

%o for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))

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

%K nonn,tabl,changed

%O 0,5

%A _Paul D. Hanna_, Jun 02 2007, Jun 06 2007