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%I #18 Jul 26 2012 13:34:55
%S 1,1,-2,-3,-1,1,-4,-2,22,49,49,27,8,1,1,-6,3,61,15,-567,-1946,-3607,
%T -4489,-4015,-2640,-1274,-441,-104,-15,-1,1,-8,12,108,-218,-1938,-834,
%U 27124,136919,393601,809873,1288950,1646268,1720788,1487263,1067345,635682,312646
%N Triangle, read by rows of n^2 terms, where row n equals the coefficients in the series reversion of the function G(y,n)-1 such that: y = Sum_{m>=1} 1/G(y,n)^(2*n*m) * Product_{k=1..m} (1 - 1/G(y,n)^(2*k-1)).
%H Paul D. Hanna, <a href="/A214690/b214690.txt">Rows n = 1..12, flattened.</a>
%F G.f.: A(x,y) = Sum_{n>=1} -x^n * Product_{k=1..n} (1 - (1+y)^(2*k-1)) / (1 - x*(1+y)^(2*k-1)).
%F G.f. for row n is R(y,n) = Sum_{k=1..n^2} y^k*T(n,k) defined by:
%F A(x,y) = Sum_{n>=1} x^n * R(y,n) such that:
%F R(y,n) = Series_Reversion( G(y,n) - 1 ) where G(y,n) satisfies:
%F y = Sum_{m>=1} 1/G(y,n)^(2*n*m) * Product_{k=1..m} (1 - 1/G(y,n)^(2*k-1)), for n>=1.
%F Row polynomials R(y,n) satisfy:
%F (1) R(1,n) = -(-1)^n * A214691(n) for n>=1.
%F (2) R(-1,n) = 1 for n>=1.
%F (3) R'(-1,n) = 0 for n>1.
%e Consider the family of power series G(x,n) that satisfy:
%e x = Sum_{m>=1} 1/G(x,n)^(2*n*m) * Product_{k=1..m} (1 - 1/G(x,n)^(2*k-1)).
%e Examples of sequences with g.f. G(x,n) are:
%e n=2: A214692 = [1, 1, 2, 11, 71, 515, 3997, 32488, 273009, ...];
%e n=3: A214693 = [1, 1, 4, 34, 338, 3691, 42623, 510949, 6289912, ...];
%e n=4: A214694 = [1, 1, 6, 69, 929, 13692, 213402, 3456450, ...];
%e n=5: A214695 = [1, 1, 8, 116, 1972, 36682, 722098, 14784834, ...]; ...
%e Observe that Series_Reversion(G(x,n) - 1) is given by the polynomials:
%e n=1: x;
%e n=2: x - 2*x^2 - 3*x^3 - x^4;
%e n=3: x - 4*x^2 - 2*x^3 + 22*x^4 + 49*x^5 + 49*x^6 + 27*x^7 + 8*x^8 + x^9;
%e n=4: x - 6*x^2 + 3*x^3 + 61*x^4 + 15*x^5 - 567*x^6 - 1946*x^7 - 3607*x^8 - 4489*x^9 - 4015*x^10 - 2640*x^11 - 1274*x^12 - 441*x^13 - 104*x^14 - 15*x^15 - x^16; ...
%e This triangle of coefficients in the above polynomials begins:
%e [1];
%e [1, -2, -3, -1];
%e [1, -4, -2, 22, 49, 49, 27, 8, 1];
%e [1, -6, 3, 61, 15, -567, -1946, -3607, -4489, -4015, -2640, -1274, -441, -104, -15, -1];
%e [1, -8, 12, 108, -218, -1938, -834, 27124, 136919, 393601, 809873, 1288950, 1646268, 1720788, 1487263, 1067345, 635682, 312646, 125761, 40734, 10373, 2001, 275, 24, 1];
%e [1, -10, 25, 155, -750, -3562, 12824, 113082, 113375, -2035735, -14707914, -59955129, -179036484, -426054391, -841492130, -1412100002, -2043288274, -2574420276, -2842741390, -2762638817, -2368603455, -1793326192, -1198603784, -706071990, -365534676, -165596757, -65259715, -22195440, -6446730, -1576815, -318649, -51799, -6511, -594, -35, -1]; ...
%o (PARI) {T(n, k)=local(Axy=x*y); Axy=sum(m=1, n, -x^m*prod(j=1, m, (1-(1+y)^(2*j-1))/(1-x*(1+y)^(2*j-1))+x*O(x^n))); polcoeff(polcoeff(Axy, n, x), k, y)}
%o {for(n=1, 10, for(k=1, n^2, print1(T(n, k), ", ")); print(""))}
%o (PARI) {a(n, p)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(2*p*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
%o {for(n=1, 8, Tn=Vec(serreverse(sum(m=1, n^2, a(m, n)*x^m)+x*O(x^(n^2)))); for(k=1, n^2, print1(Tn[k], ", ")); print(""))}
%Y Cf. A214691 (row sums), A214692, A214693, A214694, A214695, A214670 (variant).
%K sign,tabf
%O 1,3
%A _Paul D. Hanna_, Jul 25 2012
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