OFFSET
0,3
COMMENTS
In general, Sum_{k=0..n} m^k * n!/k! * binomial(m*n-k-m, n-k) * (k-1)/(n-1) is for m>1 asymptotic to m^(m*(n-1)-1/2) / (m-1)^((m-1)*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014
FORMULA
Let G(x) = 1 + x*G(x)^10 be the g.f. of A059968, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^9.
(2) A'(x) = exp(10*x*G(x)^9).
(3) A(x) = exp( Integral G(x)^9 dx ).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251590.
(6) A(x) = Sum_{n>=0} A251590(n)*(x/A(x))^n/n! and
(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251590(n),
where A251590(n) = 10^(n-8) * (n+1)^(n-10) * (4782969*n^8 + 50309748*n^7 + 237013938*n^6 + 655232760*n^5 + 1166624361*n^4 + 1374998212*n^3 + 1051760172*n^2 + 479277840*n + 100000000).
a(n) = Sum_{k=0..n} 10^k * n!/k! * binomial(10*n-k-10, n-k) * (k-1)/(n-1) for n>1.
Recurrence: 81*(3*n-5)*(3*n-4)*(9*n-17)*(9*n-16)*(9*n-14)*(9*n-13)*(9*n-11)*(9*n-10)*(250000*n^8 - 5300000*n^7 + 49332500*n^6 - 263500000*n^5 + 884055975*n^4 - 1909634570*n^3 + 2596659373*n^2 - 2035277286*n + 705468040)*a(n) = 800*(3125000000000*n^17 - 111562500000000*n^16 + 1872125000000000*n^15 - 19618187500000000*n^14 + 143829395937500000*n^13 - 783195370343750000*n^12 + 3281447638218750000*n^11 - 10810863753751875000*n^10 + 28370066880833218750*n^9 - 59681174371832246875*n^8 + 100725400409628775000*n^7 - 135736802338370325750*n^6 + 144424061701272600950*n^5 - 118936947986511839915*n^4 + 73322264536912326596*n^3 - 31942069342168467356*n^2 + 8798129066413437408*n - 1156512281566561920)*a(n-1) + 10000000000*(250000*n^8 - 3300000*n^7 + 19232500*n^6 - 64805000*n^5 + 138543475*n^4 - 193260670*n^3 + 172779013*n^2 - 91243350*n + 22054032)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 10^(10*(n-1)-1/2) / 9^(9*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 10*x^2/2! + 280*x^3/3! + 13960*x^4/4! + 1023760*x^5/5! +...
such that A(x) = exp(10*x*G(x)^9) / G(x)^9
where G(x) = 1 + x*G(x)^10 is the g.f. of A059968:
G(x) = 1 + x + 10*x^2 + 145*x^3 + 2470*x^4 + 46060*x^5 + 910252*x^6 +...
Note that
A'(x) = exp(10*x*G(x)^9) = 1 + 10*x + 280*x^2/2! + 13960*x^3/3! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 9*x^2/2 + 252*x^3/3 + 12654*x^4/4 + 933984*x^5/5 +...
and so A'(x)/A(x) = G(x)^9.
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, 10, 280, 13960, 1023760, 99935200, 12226859200, ...];
n=2: [1, 2, 22, 620, 30760, 2243120, 217911520, 26556406400, ...];
n=3: [1, 3, 36, 1026, 50760, 3683880, 356283360, 43256151360, ...];
n=4: [1, 4, 52, 1504, 74344, 5374240, 517647520, 62621962240, ...];
n=5: [1, 5, 70, 2060, 101920, 7344920, 704861200, 84980501600, ...];
n=6: [1, 6, 90, 2700, 133920, 9629280, 921060720, 110691813600, ...];
n=7: [1, 7, 112, 3430, 170800, 12263440, 1169680960, 140152067440, ...];
n=8: [1, 8, 136, 4256, 213040, 15286400, 1454475520, 173796462080, ...]; ...
in which the main diagonal begins (see A251587):
[1, 2, 36, 1504, 101920, 9629280, 1169680960, 173796462080, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 10^(n-8) * (n+1)^(n-9) * (4782969*n^8 + 50309748*n^7 + 237013938*n^6 + 655232760*n^5 + 1166624361*n^4 + 1374998212*n^3 + 1051760172*n^2 + 479277840*n + 100000000) for n>=0.
MATHEMATICA
Flatten[{1, 1, Table[Sum[10^k * n!/k! * Binomial[10*n-k-10, 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^10 +x*O(x^n)); n!*polcoeff( exp(10*x*G^9) / G^9, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = if(n==0, 1, sum(k=0, n, 10^k * n!/k! * binomial(10*n-k-10, n-k)*if(n==1, 1/10, (k-1)/(n-1)) ))}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 06 2014
STATUS
approved