OFFSET
1,4
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
EXAMPLE
G.f.: 1 = ... (((((((((exp(x) - 1*x)^2 - 1*x^2)^3 - 1*x^3)^4 - 4*x^4)^5 - 11*x^5)^6 - 66*x^6)^7 - 617*x^7)^8 - 14904*x^8)^9 - 133191*x^9)^10 -...- a(n)*x^n )^(n+1) -...
ILLUSTRATION OF GENERATING METHOD.
Start with G1 = exp(x), and proceed as follows:
G2 = (G1 - 1*x)^2 = 1 + x^2 + 1/3*x^3 + 1/3*x^4 + 11/60*x^5 + 13/180*x^6 +...
G3 = (G2 - 1*x^2)^3 = 1 + x^3 + x^4 + 11/20*x^5 + 11/20*x^6 + 617/840*x^7 +...
G4 = (G3 - 1*x^3)^4 = 1 + 4*x^4 + 11/5*x^5 + 11/5*x^6 + 617/210*x^7 + 621/70*x^8 +...
G5 = (G4 - 4*x^4)^5 = 1 + 11*x^5 + 11*x^6 + 617/42*x^7 + 621/14*x^8 +...
G6 = (G5 - 11*x^5)^6 = 1 + 66*x^6 + 617/7*x^7 + 1863/7*x^8 + 14799/56*x^9 +...
G7 = (G6 - 66*x^6)^7 = 1 + 617*x^7 + 1863*x^8 + 14799/8*x^9 + 297937/72*x^10 +...
G8 = (G7 - 617*x^7)^8 = 1 + 14904*x^8 + 14799*x^9 + 297937/9*x^10 +...
G9 = (G8 - 14904*x^8)^9 = 1 + 133191*x^9 + 297937*x^10 + 54952349/110*x^11 +...
G10 = (G9 - 133191*x^9)^10 = 1 + 2979370*x^10 + 54952349/11*x^11 +...
...
G_{n+1} = (G_{n} - a(n)*x^n)^(n+1) = 1 + a(n+1)*x^(n+1) + a(n+2)*x^(n+2)/(n+2) +...
...
Also, working backwards from the n-th term and taking roots yields exp(x) as a limit; for example, working backwards from the 9th term, we get:
((((((((1 + 133191*x^9)^(1/9) + 14904*x^8)^(1/8) + 617*x^7)^(1/7) + 66*x^6)^(1/6) + 11*x^5)^(1/5) + 4*x^4)^(1/4) + 1*x^3)^(1/3) + 1*x^2)^(1/2) + 1*x = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! + x^7/7! + x^8/8! + x^9/9! - 2979369*x^10/10! +...
PROG
(PARI) {a(n) = my(A=[1], G); for(i=1, n, A=concat(A, 0); G = exp(x +x*O(x^#A)); for(m=1, #A, G = (G - A[m]*x^m)^(m+1) ); A[#A] = polcoeff(G, #A)/(#A+1) ); A[n]}
for(n=1, 40, print1(a(n), ", "))
(PARI) /* Informal quick print of the first N terms: */
{N=100; A=[1]; G = exp(x +x^2*O(x^N)); for(m=1, N-1, A=concat(A, 0); G = (G - A[m]*x^m)^(m+1); A[m+1] = polcoeff(G, m+1); print1(A[m], ", "); ); print1(A[N], ", ") }
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 12 2016
STATUS
approved