A363137 - OEIS

%I #11 May 27 2023 00:47:45

%S 1,3,23,284,4125,65526,1102403,19305377,348217156,6425056149,

%T 120700893495,2300815588583,44391646154596,865243089927133,

%U 17011581975085968,336981451741477122,6719019528496352690,134742110298875293027,2715909284023948643846,54992586234084937679092

%N Expansion of g.f. A(x) satisfying A(x)^4 = Sum_{n=-oo..+oo} (-x)^n * (A(x)^5 + x^(n-1))^(n+1).

%C Given g.f. G(x,y) of triangle A359670, then A(x) = G(x,y=A(x)^4).

%H Paul D. Hanna, <a href="/A363137/b363137.txt">Table of n, a(n) for n = 0..250</a>

%F G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.

%F (1) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} A359670(n,k) * A(x)^(4*k).

%F (2) A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^5 + x^(n-1))^(n+1).

%F (3) A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (A(x)^5 + x^n)^n.

%F (4) x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^5*x^(n+1))^(n-1).

%F (5) x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + A(x)^5*x^(n+1))^(n+1).

%F (6) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^5 + x^(n-1))^n ].

%F (7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (A(x)^5 + x^n)^n ].

%F (8) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)^5*x^(n+1))^n ].

%F (9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x)^5 + x^n)^(n+1).

%F (10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^5*x^n)^n.

%F (11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^5*x^(n+1))^n.

%e G.f.: A(x) = 1 + 3*x + 23*x^2 + 284*x^3 + 4125*x^4 + 65526*x^5 + 1102403*x^6 + 19305377*x^7 + 348217156*x^8 + 6425056149*x^9 + ...

%e where A = A(x) may be generated from triangle A359670 as follows:

%e A(x) = 1 + x*(2 + A^4) + x^2*(4 + 6*A^4 + A^8) + x^3*(8 + 21*A^4 + 12*A^8 + A^12) + x^4*(14 + 62*A^4 + 68*A^8 + 20*A^12 + A^16) + x^5*(24 + 162*A^4 + 284*A^8 + 170*A^12 + 30*A^16 + A^20) + x^6*(40 + 384*A^4 + 998*A^8 + 970*A^12 + 360*A^16 + 42*A^20 + A^24) + x^7*(64 + 855*A^4 + 3092*A^8 + 4410*A^12 + 2720*A^16 + 679*A^20 + 56*A^24 + A^28) + x^8*(100 + 1806*A^4 + 8724*A^8 + 17172*A^12 + 15627*A^16 + 6608*A^20 + 1176*A^24 + 72*A^28 + A^32) + ... + x^n*(Sum_{k=0..n} A359670(n,k) * A(x)^(4*k)) + ...

%e RELATED SERIES.

%e A(x)^2 = 1 + 6*x + 55*x^2 + 706*x^3 + 10483*x^4 + 168866*x^5 + 2868368*x^6 + 50582368*x^7 + 917211505*x^8 + 16994216980*x^9 + ...

%e A(x)^3 = 1 + 9*x + 96*x^2 + 1293*x^3 + 19695*x^4 + 322449*x^5 + 5539013*x^6 + 98484537*x^7 + 1797074331*x^8 + 33461795117*x^9 + ...

%e A(x)^4 = 1 + 12*x + 146*x^2 + 2072*x^3 + 32463*x^4 + 541188*x^5 + 9414694*x^6 + 168962408*x^7 + 3105263987*x^8 + 58149612672*x^9 + ...

%e A(x)^5 = 1 + 15*x + 205*x^2 + 3070*x^3 + 49570*x^4 + 842723*x^5 + 14864320*x^6 + 269521315*x^7 + 4992898830*x^8 + 94091310230*x^9 + ...

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

%o A[#A] = polcoeff(1 - sum(n=-#A, #A, (-1)^n * x^n * (Ser(A)^5 + x^(n-1))^(n+1) )/Ser(A)^4, #A-1, x) ); A[n+1]}

%o for(n=0, 25, print1( a(n), ", "))

%o (PARI) {a(n) = my(A=1); for(i=1, n,

%o A = 1/sum(m=-#A, #A, (-1)^m * (x*A^5 + x^m + x*O(x^n) )^m ) );

%o polcoeff( A, n, x)}

%o for(n=0, 25, print1( a(n), ", "))

%Y Cf. A361770, A363135, A363136.

%Y Cf. A359670.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 26 2023