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A363136
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Expansion of g.f. A(x) satisfying A(x)^3 = Sum_{n=-oo..+oo} (-x)^n * (A(x)^4 + x^(n-1))^(n+1).
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5
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1, 3, 20, 201, 2364, 30356, 412223, 5821790, 84640367, 1258323895, 19041449659, 292322012264, 4541588520144, 71272574697572, 1128153098245655, 17990251268286993, 288748431461195204, 4660994483707782316, 75619617951059214712, 1232402387922242020729
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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)^3).
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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)^(3*k).
(2) A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^4 + x^(n-1))^(n+1).
(3) A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (A(x)^4 + x^n)^n.
(4) x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^4*x^(n+1))^(n-1).
(5) x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + A(x)^4*x^(n+1))^(n+1).
(6) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^4 + x^(n-1))^n ].
(7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (A(x)^4 + x^n)^n ].
(8) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)^4*x^(n+1))^n ].
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x)^4 + x^n)^(n+1).
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^4*x^n)^n.
(11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^4*x^(n+1))^n.
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EXAMPLE
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G.f.: A(x) = 1 + 3*x + 20*x^2 + 201*x^3 + 2364*x^4 + 30356*x^5 + 412223*x^6 + 5821790*x^7 + 84640367*x^8 + 1258323895*x^9 + 19041449659*x^10 + ...
where A = A(x) may be generated from triangle A359670 as follows:
A(x) = 1 + x*(2 + A^3) + x^2*(4 + 6*A^3 + A^6) + x^3*(8 + 21*A^3 + 12*A^6 + A^9) + x^4*(14 + 62*A^3 + 68*A^6 + 20*A^9 + A^12) + x^5*(24 + 162*A^3 + 284*A^6 + 170*A^9 + 30*A^12 + A^15) + x^6*(40 + 384*A^3 + 998*A^6 + 970*A^9 + 360*A^12 + 42*A^15 + A^18) + x^7*(64 + 855*A^3 + 3092*A^6 + 4410*A^9 + 2720*A^12 + 679*A^15 + 56*A^18 + A^21) + x^8*(100 + 1806*A^3 + 8724*A^6 + 17172*A^9 + 15627*A^12 + 6608*A^15 + 1176*A^18 + 72*A^21 + A^24) + ... + x^n*(Sum_{k=0..n} A359670(n,k) * A(x)^(3*k)) + ...
RELATED SERIES.
A(x)^2 = 1 + 6*x + 49*x^2 + 522*x^3 + 6334*x^4 + 82936*x^5 + 1141543*x^6 + 16281486*x^7 + 238492002*x^8 + 3566598406*x^9 + ...
A(x)^3 = 1 + 9*x + 87*x^2 + 990*x^3 + 12450*x^4 + 166767*x^5 + 2332148*x^6 + 33654549*x^7 + 497426898*x^8 + 7492104096*x^9 + ...
A(x)^4 = 1 + 12*x + 134*x^2 + 1632*x^3 + 21333*x^4 + 293036*x^5 + 4171534*x^6 + 61001912*x^7 + 910937790*x^8 + 13833090604*x^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)^4 + x^(n-1))^(n+1) )/Ser(A)^3, #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^4 + 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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