A251575 E.g.f.: exp(5*x*G(x)^4) / G(x)^4 where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.

1, 1, 5, 65, 1505, 51505, 2354725, 135258625, 9373203425, 761486105825, 71001537157925, 7475144493546625, 877222642396170625, 113551974107296500625, 16073867927431440597125, 2470217878902686107522625, 409596824402404827033730625, 72890993386914239524503090625, 13857243751694786173837746653125

Offset: 0

Author

Paul D. Hanna, Dec 06 2014

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 65*x^3/3! + 1505*x^4/4! + 51505*x^5/5! +...
such that A(x) = exp(5*x*G(x)^4) / G(x)^4
where G(x) = 1 + x*G(x)^5 is the g.f. of A002294:
G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 +...
Note that
A'(x) = exp(5*x*G(x)^4) = 1 + 5*x + 65*x^2/2! + 1505*x^3/3! + 51505*x^4/4! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 4*x^2/2 + 26*x^3/3 + 204*x^4/4 + 1771*x^5/5 +...
and so A'(x)/A(x) = G(x)^4.
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,  5,   65,  1505,   51505,  2354725,  135258625, ...];
n=2: [1, 2, 12,  160,  3680,  124560,  5637760,  321147200, ...];
n=3: [1, 3, 21,  291,  6705,  225315, 10112805,  571694355, ...];
n=4: [1, 4, 32,  464, 10784,  361120, 16101760,  904145920, ...];
n=5: [1, 5, 45,  685, 16145,  540645, 23993725, 1339552925, ...];
n=6: [1, 6, 60,  960, 23040,  774000, 34254720, 1903435200, ...];
n=7: [1, 7, 77, 1295, 31745, 1072855, 47438125, 2626525615, ...];
n=8: [1, 8, 96, 1696, 42560, 1450560, 64195840, 3545600000, ...]; ...
in which the main diagonal begins (see A251585):
[1, 2, 21, 464, 16145, 774000, 47438125, 3545600000, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 5^(n-3) * (n+1)^(n-4) * (16*n^3 + 87*n^2 + 172*n + 125) for n>=0.
		

Crossrefs

Cf. A251585, A251665, A002294, A118971.

Cf. Variants: A243953, A251573, A251574, A251576, A251577, A251578, A251579, A251580.

Programs

Mathematica

Flatten[{1,1,Table[Sum[5^k * n!/k! * Binomial[5*n-k-5, n-k] * (k-1)/(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)

PARI

{a(n) = local(G=1);for(i=1,n,G=1+x*G^5 +x*O(x^n)); n!*polcoeff(exp(5*x*G^4)/G^4, n)}
for(n=0, 20, print1(a(n), ", "))

PARI

{a(n) = if(n==0||n==1, 1, sum(k=0, n, 5^k * n!/k! * binomial(5*n-k-5,n-k) * (k-1)/(n-1) ))}
for(n=0, 20, print1(a(n), ", "))

Formula

Let G(x) = 1 + x*G(x)^5 be the g.f. of A002294, then the e.g.f. A(x) of this sequence satisfies:

(1) A'(x)/A(x) = G(x)^4.

(2) A'(x) = exp(5*x*G(x)^4).

(3) A(x) = exp( Integral G(x)^4 dx ).

(4) A(x) = exp( Sum_{n>=1} A118971(n-1)*x^n/n ), where A118971(n-1) = binomial(5*n-2,n)/(4*n-1).

(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251585.

(6) A(x) = Sum_{n>=0} A251585(n)*(x/A(x))^n/n! and

(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251585(n)

where A251585(n) = 5^(n-3) * (n+1)^(n-5) * (16*n^3 + 87*n^2 + 172*n + 125).

a(n) = Sum_{k=0..n} 5^k * n!/k! * binomial(5*n-k-5, n-k) * (k-1)/(n-1) for n>1.

Recurrence: 8*(2*n-3)*(4*n-7)*(4*n-5)*(25*n^3 - 210*n^2 + 598*n - 581)*a(n) = 5*(15625*n^7 - 240625*n^6 + 1592500*n^5 - 5883125*n^4 + 13135350*n^3 - 17781015*n^2 + 13566657*n - 4523904)*a(n-1) - 3125*(25*n^3 - 135*n^2 + 253*n - 168)*a(n-2). - Vaclav Kotesovec, Dec 07 2014

a(n) ~ 5^(5*n-11/2) * n^(n-2) / (2^(8*n-9) * exp(n-1)). - Vaclav Kotesovec, Dec 07 2014