OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) satisfies:
(1) [x^n] A(x)^(n+1) = (n+1) * [x^(n-1)] A(x)^(n+2) for n>=1.
(2) A(x) = x/Series_Reversion(G(x)) where G(x) = x + d/dx G(x)^3/3 is the g.f. of A182304; thus, G(x) = A(G(x)).
a(n) ~ c * n! * n^4, where c = 0.005428317993266202636748034138... (same as for A177384). - Vaclav Kotesovec, Jul 28 2014
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 103*x^4 + 879*x^5 + 8799*x^6 +...
The table of coefficients of x^k in A(x)^n begin:
n=1: [1, 1, 3, 15, 103, 879, 8799, 99711, 1250175, ...]
n=2: [1, 2, 7, 36, 245, 2054, 20199, 225384, 2789545, ...]
n=3: [1, 3, 12, 64, 435, 3597, 34806, 382590, 4674405, ...]
n=4: [1, 4, 18, 100, 683, 5592, 53340, 577960, 6971325, ...]
n=5: [1, 5, 25, 145, 1000, 8136, 76650, 819350, 9758835, ...]
n=6: [1, 6, 33, 200, 1398, 11340, 105730, 1116024, 13129335, ...]
n=7: [1, 7, 42, 266, 1890, 15330, 141736, 1478856, 17191251, ...]
n=8: [1, 8, 52, 344, 2490, 20248, 186004, 1920552, 22071459, ...]
n=9: [1, 9, 63, 435, 3213, 26253, 240069, 2455893, 27918000, ...]
n=10:[1, 10, 75, 540, 4075, 33522, 305685, 3102000, 34903110, ...]; ...
in which the diagonals illustrate the relation
[x^n] A(x)^(n+1) = (n+1) * [x^(n-1)] A(x)^(n+2) for n>=1
as follows:
[x^1] A(x)^2 = 2 = 2*[x^0] A(x)^3 = 2*1 ;
[x^2] A(x)^3 = 12 = 3*[x^1] A(x)^4 = 3*4 ;
[x^3] A(x)^4 = 100 = 4*[x^2] A(x)^5 = 4*25 ;
[x^4] A(x)^5 = 1000 = 5*[x^3] A(x)^6 = 5*200 ;
[x^5] A(x)^6 = 11340 = 6*[x^4] A(x)^7 = 6*1890 ;
[x^6] A(x)^7 = 141736 = 7*[x^5] A(x)^8 = 7*20248 ;
[x^7] A(x)^8 = 1920552 = 8*[x^6] A(x)^9 = 8*240069 ;
[x^8] A(x)^9 = 27918000 = 9*[x^7] A(x)^10 = 9*3102000 ; ...
RELATED SERIES.
Also, from the above table we can generate A182304:
[1/1, 2/2, 12/3, 100/4, 1000/5, 11340/6, 141736/7, 1920552/8, ...]
= [1, 1, 4, 25, 200, 1890, 20248, 240069, 3102000, 43226590, ...];
the g.f. of A182304 begins:
G(x) = x + x^2 + 4*x^3 + 25*x^4 + 200*x^5 + 1890*x^6 + 20248*x^7 +...
where G(x) = x + d/dx G(x)^3/3 and G(x) = A(G(x)).
PROG
(PARI) /* From [x^n] A(x)^(n+1) = (n+1) * [x^(n-1)] A(x)^(n+2): */
{a(n)=local(A=[1, 1]); for(i=2, n, A=concat(A, 0); A[#A]=((#A)*Vec(Ser(A)^(#A+1))[#A-1]-Vec(Ser(A)^(#A))[#A])/(#A)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From A(x) = 1 + x*A(x)^3 / (A(x) - x*A'(x)): */
{a(n)=local(A=1+x); for(i=1, n, A = 1 + x*A^3/(A - x*A' +x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 27 2014
STATUS
approved