A366730 - OEIS

%I #7 Oct 29 2023 22:02:10

%S 1,0,1,0,-2,2,0,3,-6,5,0,-6,14,-20,14,0,11,-36,59,-70,42,0,-18,87,

%T -176,246,-252,132,0,28,-190,500,-824,1022,-924,429,0,-44,386,-1312,

%U 2615,-3780,4236,-3432,1430,0,69,-756,3218,-7734,13107,-17112,17523,-12870,4862,0,-104,1443,-7514,21496,-42444,64031,-76692,72358,-48620,16796

%N Expansion of g.f. A(x,y) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x,y)^n * (y - x^(n-1))^(n+1), as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y), read by rows n >= 0.

%H Paul D. Hanna, <a href="/A366730/b366730.txt">Table of n, a(n) for n = 0..1274</a>

%F G.f. A(x,y) = Sum_{n>=0} sum_{k=0..n} T(n,k)*x^n*y^k satisfies the following formulas.

%F (1) 0 = Sum_{n=-oo..+oo} x^n * A(x,y)^n * (y - x^(n-1))^(n+1).

%F (2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( A(x,y)^n * (1 - y*x^(n+1))^(n-1) ).

%e G.f.: A(x,y) = 1 + x*y + x^2*(-2*y + 2*y^2) + x^3*(3*y - 6*y^2 + 5*y^3) + x^4*(-6*y + 14*y^2 - 20*y^3 + 14*y^4) + x^5*(11*y - 36*y^2 + 59*y^3 - 70*y^4 + 42*y^5) + x^6*(-18*y + 87*y^2 - 176*y^3 + 246*y^4 - 252*y^5 + 132*y^6) + x^7*(28*y - 190*y^2 + 500*y^3 - 824*y^4 + 1022*y^5 - 924*y^6 + 429*y^7) + x^8*(-44*y + 386*y^2 - 1312*y^3 + 2615*y^4 - 3780*y^5 + 4236*y^6 - 3432*y^7 + 1430*y^8) + x^9*(69*y - 756*y^2 + 3218*y^3 - 7734*y^4 + 13107*y^5 - 17112*y^6 + 17523*y^7 - 12870*y^8 + 4862*y^9) + ...

%e where A = A(x,y) satisfies

%e 0 = Sum_{n=-oo..+oo} x^n * A^n * (y - x^(n-1))^(n+1);

%e explicitly,

%e 0 = ((-A + 1)/A)/x + y + (A*y^2 - 2*A*y + ((A^3 - 1)/A^2))*x + A^2*y^3*x^2 + (A^3*y^4 - 3*A^2*y^2)*x^3 + (A^4*y^5 + ((3*A^4 - 1)/A^2)*y)*x^4 + (A^5*y^6 - 4*A^3*y^3 + ((-A^5 + 1)/A^3))*x^5 + A^6*y^7*x^6 + (A^7*y^8 - 5*A^4*y^4 + ((6*A^5 - 1)/A^2)*y^2)*x^7 + A^8*y^9*x^8 + (A^9*y^10 - 6*A^5*y^5 + ((-4*A^6 + 2)/A^3)*y)*x^9 + (A^10*y^11 + ((10*A^6 - 1)/A^2)*y^3)*x^10 + ...

%e This triangle of coefficients of x^n*y^k in A(x,y) begins:

%e 1;

%e 0, 1;

%e 0, -2, 2;

%e 0, 3, -6, 5;

%e 0, -6, 14, -20, 14;

%e 0, 11, -36, 59, -70, 42;

%e 0, -18, 87, -176, 246, -252, 132;

%e 0, 28, -190, 500, -824, 1022, -924, 429;

%e 0, -44, 386, -1312, 2615, -3780, 4236, -3432, 1430;

%e 0, 69, -756, 3218, -7734, 13107, -17112, 17523, -12870, 4862;

%e 0, -104, 1443, -7514, 21496, -42444, 64031, -76692, 72358, -48620, 16796;

%e 0, 152, -2668, 16862, -56856, 129425, -223458, 307189, -340912, 298298, -184756, 58786;

%e 0, -222, 4782, -36456, 144159, -375618, 734310, -1143924, 1453221, -1504932, 1227876, -705432, 208012; ...

%e in which the main diagonal equals the Catalan numbers (A000108), and column 1 equals the coefficients in Product_{n>=1} (1 - q^(2*n-1))^2/(1 - q^(2*n))^2 (A274621).

%o (PARI) {T(n,k) = my(A=[1]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoeff( sum(n=-#A,#A, x^n * Ser(A)^n * (y - x^(n-1))^(n+1) ), #A-2)); polcoeff(A[n+1],k)}

%o for(n=0,12, for(k=0,n, print1(T(n,k),", "));print(""))

%Y Cf. A274621 (column 1), A000108 (diagonal), A366736 (central terms).

%Y Cf. A366731 (y=1), A366732 (y=2), A366733 (y=3), A366734 (y=4), A366735 (y=-1).

%K sign,tabl

%O 0,5

%A _Paul D. Hanna_, Oct 29 2023