A245312 G.f. A(x) satisfies: [x^(n+1)] A(x)^n = n^2 * ( [x^(n-1)] A(x)^n ) for n>=1.

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

Paul D. Hanna, Table of n, a(n) for n = 0..400

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

Cf. A245313, A245310.

Sequence in context: A181077 A158873 A103591 * A018932 A111562 A009654

Adjacent sequences: A245309 A245310 A245311 * A245313 A245314 A245315

Keyword

nonn

Author

Paul D. Hanna, Jul 19 2014

Status

approved