A137570 - OEIS

%I #11 Jan 31 2025 03:49:42

%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,

%T 186,100,31,7,1,1,10,53,230,397,150,40,8,1,1,11,63,283,1281,681,211,

%U 50,9,1,1,12,74,346,1564,2802,1051,284,61,10,1,1,13,86,420,1910,9294,4908

%N 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.

%F G.f.: A(x,y) = D(x*y)*(1/(1 - y*F(x*y)) + x*C(x*y)*F(x*y)^2/(1 - x*C(x*y))), where D(x) = 1/(1 - x*C(x)*F(x)^2 - x*F(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.

%e Square array begins:

%e   (1),(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, 5,(10),(16), 23, 31, 40, 50, 61, 73, 86, 100, 115, 131, 148, ...;

%e   1, 6, 29,(60),(100), 150, 211, 284, 370, 470, 585, 716, 864, ...;

%e   1, 7, 36, 186,(397),(681), 1051, 1521, 2106, 2822, 3686, 4716, ...;

%e   1, 8, 44, 230, 1281,(2802),(4908), 7730, 11416, 16132, 22063, ...;

%e   1, 9, 53, 283, 1564, 9294,(20710),(36842), 58905, 88319, 126730, ...;

%e   1, 10, 63, 346, 1910, 11204, 70109,(158428),(285158), 461190, ...;

%e   1, 11, 74, 420, 2330, 13534, 83643, 544833,(1244413),(2260257), ...;

%e   ...

%e For each row, remove the terms along the diagonals (in parenthesis),

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

%e GENERATING FUNCTIONS.

%e The g.f. of n-th lower diagonal equals D(x)*F(x)^2*C(x)^n and

%e the g.f. of n-th upper diagonal equals D(x)*F(x)^n,

%e where D(x) is g.f. of main diagonal (A137571):

%e [1, 2, 10, 60, 397, 2802, 20710, 158428, 1244413, 9980220, ...]

%e defined by:

%e D(x) = 1/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where

%e C(x) = 1 + x*C(x)^2 is g.f. of Catalan numbers (A000108):

%e [1, 1, 2, 5, 14, 42, 132, 429, 1430, ..., C(2*n,n)/(n+1), ...] and

%e F(x) = 1 + x*F(x)^4 is g.f. of A002293:

%e [1, 1, 4, 22, 140, 969, 7084, 53820, ..., C(4*n,n)/(3*n+1), ...].

%o (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))))

%o (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)

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

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Jan 27 2008