login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A245312
G.f. A(x) satisfies: [x^(n+1)] A(x)^n = n^2 * ( [x^(n-1)] A(x)^n ) for n>=1.
3
1, 1, 1, 3, 10, 60, 360, 2940, 24528, 247968, 2595920, 31175496, 389671200, 5422095536, 78605082528, 1244958773760, 20527083114496, 364984417934400, 6745106725383168, 133136189132775360, 2726068542132666240, 59173740044950124160, 1329834118793805335040, 31493916740885086274304
OFFSET
0,4
COMMENTS
Do the following limits exist? If so, what are the respective values?
(1) limit a(n)*sqrt(n+1)/(n+1)! ? (Value is near 0.263981 at n=400.)
(2) limit A245313(n)/A245312(n) ? (Value is near 2.721747 at n=400.)
Limit a(n)*sqrt(n+1)/(n+1)! = 0.264485471610277836304036..., limit A245313(n)/A245312(n) = e. - Vaclav Kotesovec, Jul 20 2014
LINKS
FORMULA
Let A = A(x), the g.f. of this sequence, and G(x) = g.f. of A245313, then
(1) A(x) = 1 + x + x^2 + 3*x^3*A'/(A - x*A') + 2*x^4*(A')^2/(A - x*A')^2 + x^4*A^2*A''/(A - x*A')^3.
(2) A(x) = 1 + x + x^2 + 3*x^3*G'(x/A)/A^2 + x^4*G''(x/A)/A^3.
(3) G'(x/A) = A^2*A'/(A - x*A').
(4) G''(x/A) = 2*A^3*(A')^2/(A - x*A')^2 + A^5*A''/(A - x*A')^3.
(5) A(x) = G(x/A(x)) and G(x) = A(x*G(x)).
(6) A(x) = x / Series_Reversion(x*G(x)).
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 60*x^5 + 360*x^6 +...
The table of coefficients of x^k in A(x)^n begin:
n=1: [1, 1, 1, 3, 10, 60, 360, 2940, 24528, 247968, ...];
n=2: [1, 2, 3, 8, 27, 146, 869, 6780, 56116, 554232, ...];
n=3: [1, 3, 6, 16, 54, 270, 1576, 11796, 96441, 931539, ...];
n=4: [1, 4, 10, 28, 95, 448, 2548, 18344, 147631, 1395396, ...];
n=5: [1, 5, 15, 45, 155, 701, 3875, 26885, 212385, 1964755, ...];
n=6: [1, 6, 21, 68, 240, 1056, 5676, 38016, 294126, 2662868, ...];
n=7: [1, 7, 28, 98, 357, 1547, 8106, 52508, 397194, 3518354, ...];
n=8: [1, 8, 36, 136, 514, 2216, 11364, 71352, 527087, 4566528, ...];
n=9: [1, 9, 45, 183, 720, 3114, 15702, 95814, 690759, 5851051, 55951479]; ...
from which we can illustrate [x^(n+1)] A(x)^n = n^2*([x^(n-1)] A(x)^n):
n=1: [x^2] A(x) = 1 = 1*([x^0] A(x)) = 1*1 ;
n=2: [x^3] A(x)^2 = 8 = 2^2*([x^1] A(x)^2) = 2^2*2 ;
n=3: [x^4] A(x)^3 = 54 = 3^2*([x^2] A(x)^3) = 3^2*6 ;
n=4: [x^5] A(x)^4 = 448 = 4^2*([x^3] A(x)^4) = 4^2*28 ;
n=5: [x^6] A(x)^5 = 3875 = 5^2*([x^4] A(x)^5) = 5^2*155 ;
n=6: [x^7] A(x)^6 = 38016 = 6^2*([x^5] A(x)^6) = 6^2*1056 ;
n=7: [x^8] A(x)^7 = 397194 = 7^2*([x^6] A(x)^7) = 7^2*8106 ;
n=8: [x^9] A(x)^8 = 4566528 = 8^2*([x^7] A(x)^8) = 8^2*71352 ;
n=9: [x^10] A(x)^9 = 55951479 = 9^2*([x^8] A(x)^9) = 9^2*690759 ; ...
describing terms that lie along diagonals in the above table.
From the main diagonal in the above table, we may derive A245313:
[1/1, 2/2, 6/3, 28/4, 155/5, 1056/6, 8106/7, 71352/8, 690759/9, ...]
= [1, 1, 2, 7, 31, 176, 1158, 8919, 76751, ...].
PROG
(PARI) /* From: [x^(n-1)] A(x)^n = n^2*([x^(n+1)] A(x)^n) */
{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); m=#A-2; A[#A]=-Vec(Ser(A)^m*(1-m^2*x^2))[#A]/m); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From differential equation: */
{a(n)=local(A=1+x, D=1); for(i=1, n, D=(A - x*A'+x*O(x^n));
A=1+x+x^2 + 3*x^3*A'/D + 2*x^4*(A')^2/D^2 + x^4*A^2*A''/D^3); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Sequence in context: A181077 A158873 A103591 * A018932 A111562 A009654
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 19 2014
STATUS
approved