A249786 - OEIS

%I #32 Nov 15 2014 10:17:11

%S 1,1,-2,6,-48,360,-4800,58800,-1088640,18627840,-440294400,9699782400,

%T -278672486400,7519473561600,-254211897139200,8123999659776000,

%U -315817889587200000,11668326078689280000,-512656874530504704000,21503534793369108480000,-1053509824992697712640000

%N E.g.f. A(x) satisfies: (A(x)^2 - 4*x)^3 = (2 - A(x)^3)^2.

%H Vaclav Kotesovec, <a href="/A249786/b249786.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f.: (1 + 3*Series_Reversion(G(x)))^(1/3), where G(x) = ((1+3*x)^(2/3) - (1-3*x)^(2/3))/4  = x + Sum_{n>=1} x^(2*n+1)/(2*n+1)! * Product_{k=0..n-1} (6*k+1)*(6*k+4).

%F E.g.f. A(x) satisfies:

%F (1) A(x)^3 + A(-x)^3 = 2.

%F (2) A(x)^2 - A(-x)^2 = 4*x.

%F (3) x = (A(x)^2 - (2 - A(x)^3)^(2/3))/4.

%F a(n) ~ (-1)^(n+1) * 2^(4*n/3-1/6) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Nov 15 2014

%e E.g.f.: A(x) = 1 + x - 2*x^2/2! + 6*x^3/3! - 48*x^4/4! + 360*x^5/5! - 4800*x^6/6! + 58800*x^7/7! - 1088640*x^8/8! + 18627840*x^9/9! - 440294400*x^10/10! +...

%e where

%e A(x)^2 = 1 + 2*x - 2*x^2/2! - 24*x^4/4! - 1680*x^6/6! - 295680*x^8/8! - 97977600*x^10/10! - 52583731200*x^12/12! - 41661536716800*x^14/14! +...

%e A(x)^3 = 1 + 3*x - 12*x^3/3! - 360*x^5/5! - 40320*x^7/7! - 9797760*x^9/9! - 4151347200*x^11/11! - 2717056742400*x^13/13! - 2542118971392000*x^15/15! +...

%e Thus the coefficients of odd powers of x in A(x)^2 equal zero:

%e [1, 2, -2, 0, -24, 0, -1680, 0, -295680, 0, -97977600, 0, -52583731200, 0,...],

%e while the coefficients of even powers of x in A(x)^3 equal zero:

%e [1, 3, 0, -12, 0, -360, 0, -40320, 0, -9797760, 0, -4151347200, 0, ...],

%e after a few initial terms.

%e EXPLICIT FORMULA.

%e Let G(x) = ((1+3*x)^(2/3) - (1-3*x)^(2/3))/4, which begins

%e G(x) = x + 4*x^3/3! + 4*70*x^5/5! + 4*70*208*x^7/7! + 4*70*208*418*x^9/9! + 4*70*208*418*700*x^11/11! +...+ [Product_{k=0..n-1} (6*k+1)*(6*k+4)]*x^(2*n+1)/(2*n+1)! +...

%e then (A(x)^3 - 1)/3 = Series_Reversion(G(x)).

%e The coefficients in G(x) form triple factorials (A007559) that begin:

%e [1, 0, 4, 0, 280, 0, 58240, 0, 24344320, 0, 17041024000, 0, ...].

%o (PARI) /* Explicit formula: */

%o {a(n)=local(A,X=x+x^2*O(x^n), G=((1+3*X)^(2/3) - (1-3*X)^(2/3))/4);

%o A=(1 + 3*serreverse(G))^(1/3); n!*polcoeff(A, n)}

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

%o (PARI) /* Formula using series expansion: */

%o {a(n)=local(A, G=x + sum(m=1, n\2+1, x^(2*m+1)/(2*m+1)!*prod(k=0, m-1, (6*k+1)*(6*k+4)) +x^2*O(x^n)));

%o A=(1 + 3*serreverse(G))^(1/3); n!*polcoeff(A, n)}

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

%o (PARI) /* Alternating zero coefficients in A(x)^2 and A(x)^3: */

%o {a(n)=local(A=[1,1],E=1,M);for(i=1,n,A=concat(A,0);M=#A;

%o E=sum(m=0,M-1,A[m+1]*x^m/m!)+x*O(x^M);

%o A[M]=if(M%2==0,-(M-1)!*Vec(E^2/2)[M],-(M-1)!*Vec(E^3/3)[M]));A[n+1]}

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

%Y Cf. A249785 (dual), A249787, A249788, A249789.

%K sign

%O 0,3

%A _Paul D. Hanna_, Nov 13 2014