OFFSET
0,3
FORMULA
Let G(x) = 1 + x*G(x)^6 be the g.f. of A002295, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^5.
(2) A'(x) = exp(6*x*G(x)^5).
(3) A(x) = exp( Integral G(x)^5 dx ).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251586.
(6) A(x) = Sum_{n>=0} A251586(n)*(x/A(x))^n/n! and
(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251586(n),
where A251586(n) = 6^(n-4) * (n+1)^(n-6) * (125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296).
a(n) = Sum_{k=0..n} 6^k * n!/k! * binomial(6*n-k-6, n-k) * (k-1)/(n-1) for n>1.
Recurrence: 5*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(9*n^4 - 99*n^3 + 413*n^2 - 777*n + 559)*a(n) = 72*(5832*n^9 - 113724*n^8 + 986580*n^7 - 5003586*n^6 + 16373448*n^5 - 35916483*n^4 + 52931854*n^3 - 50678109*n^2 + 28701206*n - 7357350)*a(n-1) + 46656*(9*n^4 - 63*n^3 + 170*n^2 - 212*n + 105)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 6^(6*(n-1)-1/2) / 5^(5*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 6*x^2/2! + 96*x^3/3! + 2736*x^4/4! + 115056*x^5/5! +...
such that A(x) = exp(6*x*G(x)^5) / G(x)^5
where G(x) = 1 + x*G(x)^6 is the g.f. of A002295:
G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...
Note that
A'(x) = exp(6*x*G(x)^5) = 1 + 6*x + 96*x^2/2! + 2736*x^3/3! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 5*x^2/2 + 40*x^3/3 + 385*x^4/4 + 4095*x^5/5 + 46376*x^6/6 +...
and so A'(x)/A(x) = G(x)^5.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1, 6, 96, 2736, 115056, 6455376, 454666176, ...];
n=2: [1, 2, 14, 228, 6456, 268992, 14968224, 1047087648, ...];
n=3: [1, 3, 24, 402, 11376, 470808, 26011584, 1808151552, ...];
n=4: [1, 4, 36, 624, 17736, 730944, 40143456, 2774490624, ...];
n=5: [1, 5, 50, 900, 25800, 1061400, 58017600, 3989340000, ...];
n=6: [1, 6, 66, 1236, 35856, 1475856, 80395056, 5503484736, ...];
n=7: [1, 7, 84, 1638, 48216, 1989792, 108156384, 7376303088, ...];
n=8: [1, 8, 104, 2112, 63216, 2620608, 142314624, 9676910592, ...]; ...
in which the main diagonal begins (see A251586):
[1, 2, 24, 624, 25800, 1475856, 108156384, 9676910592, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 6^(n-4) * (n+1)^(n-5) * (125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296) for n>=0.
MATHEMATICA
Flatten[{1, 1, Table[Sum[6^k * n!/k! * Binomial[6*n-k-6, n-k] * (k-1)/(n-1), {k, 0, n}], {n, 2, 20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)
PROG
(PARI) {a(n) = local(G=1); for(i=1, n, G=1+x*G^6 +x*O(x^n)); n!*polcoeff(exp(6*x*G^5)/G^5, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = if(n==0|n==1, 1, sum(k=0, n, 6^k * n!/k! * binomial(6*n-k-6, n-k) * (k-1)/(n-1) ))}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 06 2014
STATUS
approved