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A361770
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Expansion of g.f. A(x) satisfying A(x) = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^2 + x^(n-1))^(n+1).
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5
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1, 3, 14, 80, 510, 3498, 25145, 186972, 1426159, 11096944, 87736474, 702837098, 5692337206, 46533458472, 383450469145, 3181746494524, 26562082580277, 222941953595054, 1880174585677589, 15924467403391355, 135396623401761765, 1155230973031795808, 9888061401816818319
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OFFSET
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0,2
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COMMENTS
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Given g.f. G(x,y) of triangle A359670, then A(x) = G(x,y=A(x)).
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
(1) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} A359670(n,k) * A(x)^k.
(2) A(x) = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^2 + x^(n-1))^(n+1).
(3) A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (A(x)^2 + x^n)^n.
(4) x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^2*x^(n+1))^(n-1).
(5) x*A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + A(x)^2*x^(n+1))^(n+1).
(6) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^2 + x^(n-1))^n ].
(7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (A(x)^2 + x^n)^n ].
(8) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)^2*x^(n+1))^n ].
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x)^2 + x^n)^(n+1).
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^2*x^n)^n.
(11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^2*x^(n+1))^n.
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EXAMPLE
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G.f.: A(x) = 1 + 3*x + 14*x^2 + 80*x^3 + 510*x^4 + 3498*x^5 + 25145*x^6 + 186972*x^7 + 1426159*x^8 + 11096944*x^9 + 87736474*x^10 + ...
where A = A(x) may be generated from triangle A359670 as follows:
A(x) = 1 + x*(2 + A) + x^2*(4 + 6*A + A^2) + x^3*(8 + 21*A + 12*A^2 + A^3) + x^4*(14 + 62*A + 68*A^2 + 20*A^3 + A^4) + x^5*(24 + 162*A + 284*A^2 + 170*A^3 + 30*A^4 + A^5) + x^6*(40 + 384*A + 998*A^2 + 970*A^3 + 360*A^4 + 42*A^5 + A^6) + x^7*(64 + 855*A + 3092*A^2 + 4410*A^3 + 2720*A^4 + 679*A^5 + 56*A^6 + A^7) + x^8*(100 + 1806*A + 8724*A^2 + 17172*A^3 + 15627*A^4 + 6608*A^5 + 1176*A^6 + 72*A^7 + A^8) + ... + x^n*(Sum_{k=0..n} A359670(n,k)*A(x)^k) + ...
Also, A(x) = G(x,y=1) where G(x,y) satisfies
y*G(x,y) = Sum_{n=-oo..+oo} (-1)^n * x^n * (y*G(x,y)^2 + x^(n-1))^(n+1).
Explicitly,
G(x,y) = 1 + x*(2 + y) + x^2*(4 + 8*y + 2*y^2) + x^3*(8 + 37*y + 30*y^2 + 5*y^3) + x^4*(14 + 136*y + 234*y^2 + 112*y^3 + 14*y^4) + x^5*(24 + 432*y + 1320*y^2 + 1260*y^3 + 420*y^4 + 42*y^5) + x^6*(40 + 1232*y + 6093*y^2 + 9824*y^3 + 6240*y^4 + 1584*y^5 + 132*y^6) + x^7*(64 + 3245*y + 24402*y^2 + 60543*y^3 + 62880*y^4 + 29403*y^5 + 6006*y^6 + 429*y^7) + x^8*(100 + 8024*y + 87754*y^2 + 315616*y^3 + 490405*y^4 + 365816*y^5 + 134134*y^6 + 22880*y^7 + 1430*y^8) + x^9*(154 + 18832*y + 289812*y^2 + 1448744*y^3 + 3178302*y^4 + 3476418*y^5 + 1993992*y^6 + 598312*y^7 + 87516*y^8 + 4862*y^9) + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(1 - sum(n=-#A, #A, (-1)^n * x^n * (Ser(A)^2 + x^(n-1))^(n+1) )/Ser(A), #A-1, x) ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n,
A = 1/sum(m=-#A, #A, (-1)^m * (x*A^2 + x^m + x*O(x^n) )^m ) );
polcoeff( A, n, x)}
for(n=0, 25, print1( a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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