A277039 E.g..f. A(x) satisfies: 1 = ...(((( A(x) - a(2)*x )^2 - a(3)*x^2 )^3 - a(4)*x^3 )^4 - a(5)*x^4 )^5 -..., with a(0) = a(1) = 1.

1, 1, 1, 1, 1, 4, 14, 79, 758, 16865, 191965, 4642399, 86248902, 3501670714, 115114996057, 8565607353234, 490214100103707, 84938404987634353, 3664613039216888961, 479579554530728437490, 35532408630828993454198, 8604356703941292707727453, 611094051996156994790949416, 149400677551660775985467062785, 19315916074995756159830225305613

Offset

0,6

Links

Paul D. Hanna, Table of n, a(n) for n = 0..155

Example

E.g.f.: A(x) =  1 + x + x^2/2! + x^3/3! + x^4/4! + 4*x^5/5! + 14*x^6/6! + 79*x^7/7! + 758*x^8/8! + 16865*x^9/9! + 191965*x^10/10! + 4642399*x^11/11! + 86248902*x^12/12! + 3501670714*x^13/13! + 115114996057*x^14/14! + 8565607353234*x^15/15! + 490214100103707*x^16/16! +...
such that
1 = ... (((((((((( A(x) - 1*x)^2 - 1*x^2)^3 - 1*x^3)^4 - 4*x^4)^5 - 14*x^5)^6 - 79*x^6)^7 - 758*x^7)^8 - 16865*x^8)^9 - 191965*x^9)^10 - 4642399*x^10)^11 -...- a(n+1)*x^n )^(n+1) -...
Illustration of generating method.
The coefficients in the e.g.f. A(x) are set equal to the coefficients found while using the following method of nested powers.
Start with G1 = A(x), and proceeed in this way:
G2 = (G1 - x)^2
G3 = (G2 - x^2)^3
G4 = (G3 - x^3)^4
G5 = (G4 - 4*x^4)^5
G6 = (G5 - 14*x^5)^6
G7 = (G6 - 79*x^6)^7
G8 = (G7 - 758*x^7)^8
G9 = (G8 - 16865*x^8)^9
G10 = (G9 - 191965*x^9)^10
G11 = (G10 - 4642399*x^10)^11
G12 = (G11 - 86248902*x^11)^12
G13 = (G12 - 3501670714*x^12)^13
G14 = (G13 - 115114996057*x^13)^14
...
where the coefficients in the above steps are found in the resultant series:
G2 = 1 + x^2 + 1/3*x^3 + 1/3*x^4 + 7/30*x^5 + 13/120*x^6 +...
G3 = 1 + x^3 + x^4 + 7/10*x^5 + 79/120*x^6 + 379/420*x^7 +...
G4 = 1 + 4*x^4 + 14/5*x^5 + 79/30*x^6 + 379/105*x^7 +...
G5 = 1 + 14*x^5 + 79/6*x^6 + 379/21*x^7 + 16865/336*x^8 +...
G6 = 1 + 79*x^6 + 758/7*x^7 + 16865/56*x^8 + 191965/504*x^9 +...
G7 = 1 + 758*x^7 + 16865/8*x^8 + 191965/72*x^9 + 4642399/720*x^10 +...
G8 = 1 + 16865*x^8 + 191965/9*x^9 + 4642399/90*x^10 +...
G9 = 1 + 191965*x^9 + 4642399/10*x^10 + 43124451/55*x^11 +...
G10 = 1 + 4642399*x^10 + 86248902/11*x^11 + 1750835357/66*x^12 +...
G11 = 1 + 86248902*x^11 + 1750835357/6*x^12 + 115114996057/156*x^13 +...
G12 = 1 + 3501670714*x^12 + 115114996057/13*x^13 +...
...

Prog

(PARI) {a(n) = my(A=[1, 1, 1], G=1); for(i=1, n, A = concat(A, 0);
G = sum(m=0, #A-1, A[m+1]*x^m/m!) +O(x^#A);
for(k=1, #A-3, G = (G - A[k+2]*x^k)^(k+1);
A[#A] = polcoeff(G, #A-2) )); A[n+1] }
for(n=0, 30, print1(a(n), ", "))

Crossrefs

f. A274960.

Sequence in context: A161132 A186638 A187847 * A003707 A355950 A063862

Adjacent sequences: A277036 A277037 A277038 * A277040 A277041 A277042

Keyword

nonn

Author

Paul D. Hanna, Oct 01 2016

Status

approved