A274960 G.f.: 1 = ...((((exp(x) - a(1)*x )^2 - a(2)*x^2 )^3 - a(3)*x^3 )^4 - a(4)*x^4 )^5 - ..., an infinite series of nested powers.

1, 1, 1, 4, 11, 66, 617, 14904, 133191, 2979370, 54952349, 2320492164, 74986745627, 5712761120262, 335224823645025, 63532567232899696, 2156471438897202959, 241332501820895633394, 16185395872830063013829, 3614817467354231440853820, 252852056702922700194500259, 61652901373540755514187898430, 8129145662072175831707550654665, 4051124655618938732943160094475240, 507536767300258942863758603196524375

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

Cf. A274964, A274961.

Sequence in context: A000880 A006551 A353819 * A151826 A032110 A054234

Adjacent sequences: A274957 A274958 A274959 * A274961 A274962 A274963

Keyword

nonn

Author

Paul D. Hanna, Jul 12 2016

Status

approved