login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A251579 E.g.f.: exp(9*x*G(x)^8) / G(x)^8 where G(x) = 1 + x*G(x)^9 is the g.f. of A062994. 11
1, 1, 9, 225, 10017, 656289, 57255849, 6262226721, 825067217025, 127305462542913, 22527254639457801, 4498536675388410081, 1000890043482114644769, 245556248365681036646625, 65862976584851401437170217, 19174678419336874098038167329, 6022064808176665662053835550209 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..16.

FORMULA

Let G(x) = 1 + x*G(x)^9 be the g.f. of A062994, then the e.g.f. A(x) of this sequence satisfies:

(1) A'(x)/A(x) = G(x)^8.

(2) A'(x) = exp(8*x*G(x)^8).

(3) A(x) = exp( Integral G(x)^8 dx ).

(4) A(x) = exp( Sum_{n>=1} A234513(n-1)*x^n/n ), where A234513(n-1) = binomial(9*n-2,n)/(8*n-1).

(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251589.

(6) A(x) = Sum_{n>=0} A251589(n)*(x/A(x))^n/n! and

(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251589(n),

where A251589(n) = 9^(n-7) * (n+1)^(n-9) * (262144*n^7 + 2494464*n^6 + 10470208*n^5 + 25229505*n^4 + 37857568*n^3 + 35537670*n^2 + 19414368*n + 4782969).

a(n) = Sum_{k=0..n} 9^k * n!/k! * binomial(9*n-k-9, n-k) * (k-1)/(n-1) for n>1.

Recurrence: 128*(2*n-3)*(4*n-7)*(4*n-5)*(8*n-15)*(8*n-13)*(8*n-11)*(8*n-9)*(59049*n^7 - 1102248*n^6 + 8858079*n^5 - 39764115*n^4 + 107806473*n^3 - 176772075*n^2 + 162618742*n - 64907105)*a(n) = 81*(282429536481*n^15 - 8943601988565*n^14 + 132044525265870*n^13 - 1206188364304287*n^12 + 7627178203628841*n^11 - 35382975568258428*n^10 + 124478964551078775*n^9 - 338415281830783431*n^8 + 717436315214480025*n^7 - 1187215577095780764*n^6 + 1522794566607803919*n^5 - 1488866286016780047*n^4 + 1075889068341959448*n^3 - 543536112365518695*n^2 + 172059320987344825*n - 25799292366848000)*a(n-1) - 387420489*(59049*n^7 - 688905*n^6 + 3484620*n^5 - 9940725*n^4 + 17352558*n^3 - 18650247*n^2 + 11527801*n - 3203200)*a(n-2). - Vaclav Kotesovec, Dec 07 2014

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

EXAMPLE

E.g.f.: A(x) = 1 + x + 9*x^2/2! + 225*x^3/3! + 10017*x^4/4! + 656289*x^5/5! +...

such that A(x) = exp(9*x*G(x)^8) / G(x)^8

where G(x) = 1 + x*G(x)^9 is the g.f. of A062994:

G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...

Note that

A'(x) = exp(9*x*G(x)^8) = 1 + 9*x + 225*x^2/2! + 10017*x^3/3! +...

LOGARITHMIC DERIVATIVE.

The logarithm of the e.g.f. begins:

log(A(x)) = x + 8*x^2/2 + 200*x^3/3 + 8976*x^4/4 + 592368*x^5/5 +...

and so A'(x)/A(x) = G(x)^8.

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,   9,  225,  10017,   656289,  57255849,  6262226721, ...];

n=2: [1, 2,  20,  504,  22320,  1453248, 126104256, 13731880320, ...];

n=3: [1, 3,  33,  843,  37233,  2411667, 208241361, 22581193851, ...];

n=4: [1, 4,  48, 1248,  55104,  3554496, 305558784, 33002857728, ...];

n=5: [1, 5,  65, 1725,  76305,  4906965, 420159825, 45211985325, ...];

n=6: [1, 6,  84, 2280, 101232,  6496704, 554376384, 59448214656, ...];

n=7: [1, 7, 105, 2919, 130305,  8353863, 710786601, 75977951175, ...];

n=8: [1, 8, 128, 3648, 163968, 10511232, 892233216, 95096756736, ...]; ...

in which the main diagonal begins (see A251587):

[1, 2, 33, 1248, 76305, 6496704, 710786601, 95096756736, ...]

and is given by the formula:

[x^n/n!] A(x)^(n+1) = 9^(n-7) * (n+1)^(n-8) * (262144*n^7 + 2494464*n^6 + 10470208*n^5 + 25229505*n^4 + 37857568*n^3 + 35537670*n^2 + 19414368*n + 4782969) for n>=0.

MATHEMATICA

Flatten[{1, 1, Table[Sum[9^k * n!/k! * Binomial[9*n-k-9, 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^9 +x*O(x^n)); n!*polcoeff(exp(9*x*G^8)/G^8, n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n) = if(n==0|n==1, 1, sum(k=0, n, 9^k * n!/k! * binomial(9*n-k-9, n-k) * (k-1)/(n-1) ))}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A251589, A251669, A062994, A234513.

Cf. Variants: A243953, A251573, A251574, A251575, A251576, A251577, A251578, A251580.

Sequence in context: A012749 A188662 A079727 * A128492 A294971 A001818

Adjacent sequences:  A251576 A251577 A251578 * A251580 A251581 A251582

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 06 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 23 14:15 EDT 2019. Contains 322386 sequences. (Running on oeis4.)