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A366730
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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.
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7
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1, 0, 1, 0, -2, 2, 0, 3, -6, 5, 0, -6, 14, -20, 14, 0, 11, -36, 59, -70, 42, 0, -18, 87, -176, 246, -252, 132, 0, 28, -190, 500, -824, 1022, -924, 429, 0, -44, 386, -1312, 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
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OFFSET
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0,5
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LINKS
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FORMULA
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G.f. A(x,y) = Sum_{n>=0} sum_{k=0..n} T(n,k)*x^n*y^k satisfies the following formulas.
(1) 0 = Sum_{n=-oo..+oo} x^n * A(x,y)^n * (y - x^(n-1))^(n+1).
(2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( A(x,y)^n * (1 - y*x^(n+1))^(n-1) ).
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EXAMPLE
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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) + ...
where A = A(x,y) satisfies
0 = Sum_{n=-oo..+oo} x^n * A^n * (y - x^(n-1))^(n+1);
explicitly,
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 + ...
This triangle of coefficients of x^n*y^k in A(x,y) begins:
1;
0, 1;
0, -2, 2;
0, 3, -6, 5;
0, -6, 14, -20, 14;
0, 11, -36, 59, -70, 42;
0, -18, 87, -176, 246, -252, 132;
0, 28, -190, 500, -824, 1022, -924, 429;
0, -44, 386, -1312, 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;
0, 152, -2668, 16862, -56856, 129425, -223458, 307189, -340912, 298298, -184756, 58786;
0, -222, 4782, -36456, 144159, -375618, 734310, -1143924, 1453221, -1504932, 1227876, -705432, 208012; ...
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).
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PROG
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(PARI) {T(n, k) = my(A=[1]); for(i=1, n, A = concat(A, 0);
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)}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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