A293385 G.f. A(x) satisfies: A(x) = Sum_{-oo..+oo} (x - A(x)^n)^(n+1).

1, -1, 3, -8, 25, -82, 279, -974, 3471, -12582, 46258, -172092, 646667, -2450920, 9358703, -35970203, 139053789, -540332534, 2109336474, -8268739720, 32536551783, -128468095943, 508841232024, -2021249839308, 8050245467461, -32141168565845, 128617254147007, -515764830465583, 2072316922422961, -8341724250355787, 33635614440282961, -135843921900168936

Offset: 1

Author

Paul D. Hanna, Oct 16 2017

Keywords

• sign

Comments

Compare g.f. to the identities:

(1) Sum_{-oo..+oo} (x - q^n)^(n-1) = 0,

(2) Sum_{-oo..+oo} (x - q^n)^(n+1) = x * Sum_{-oo..+oo} (x - q^n)^n,

(3) Sum_{-oo..+oo} q^n * (x - q^n)^n = 0.

Sum converges at x = r = -0.23583204404501474769372258323... with A(r) = -0.549249... and diverges for |x| > |r|.

Examples

			G.f.: A(x) = x - x^2 + 3*x^3 - 8*x^4 + 25*x^5 - 82*x^6 + 279*x^7 - 974*x^8 + 3471*x^9 - 12582*x^10 + 46258*x^11 - 172092*x^12 + 646667*x^13 - 2450920*x^14 + 9358703*x^15 - 35970203*x^16 + 139053789*x^17 - 540332534*x^18 + 2109336474*x^19 - 8268739720*x^20 +...
such that
A(x) = Sum_{-oo..+oo} (x - A(x)^n)^(n+1).
Let A = A(x), then A = P + Q where
P = (x - 1) + (x - A)^2 + (x - A^2)^3 + (x - A^3)^4 + (x - A^4)^5 + (x - A^5)^6 + (x - A^6)^7 +...+ (x - A^n)^(n+1) +...
Q = 1 - A^2/(1 - x*A^2) + A^6/(1 - x*A^3)^2 - A^12/(1 - x*A^4)^3 + A^20/(1 - x*A^5)^4 - A^30/(1 - x*A^6)^5 +...+ (-1)^(n-1)*A^(n*(n-1))/(1 - x*A^n)^(n-1) +...
More explicitly,
P = -1 + x + x^3 - x^4 + 4*x^5 - 12*x^6 + 41*x^7 - 142*x^8 + 512*x^9 - 1902*x^10 + 7227*x^11 - 27894*x^12 + 108862*x^13 - 428339*x^14 + 1695921*x^15 - 6747557*x^16 + 26951825*x^17 - 107999225*x^18 + 433924759*x^19 - 1747421503*x^20 + 7050821323*x^21 +...
Q = 1 - x^2 + 2*x^3 - 7*x^4 + 21*x^5 - 70*x^6 + 238*x^7 - 832*x^8 + 2959*x^9 - 10680*x^10 + 39031*x^11 - 144198*x^12 + 537805*x^13 - 2022581*x^14 + 7662782*x^15 - 29222646*x^16 + 112101964*x^17 - 432333309*x^18 + 1675411715*x^19 - 6521318217*x^20 + 25485730460*x^21 +...
where g.f. A(x) = P + Q.
Let A = A(x), then we can also write
A(x) = x*(1 - 2*A + 2*A^4 - 2*A^9 + 2*A^16 - 2*A^25 +...)
+ x^2*(1 - 3*A^2 + 5*A^6 - 7*A^12 + 9*A^20 - 11*A^30 +...)
+ x^3*(1 - 4*A^3 + 9*A^8 - 16*A^15 + 25*A^24 - 36*A^35 +...)
+ x^4*(1 - 5*A^4 + 14*A^10 - 30*A^18 + 55*A^28 - 91*A^40 +...)
+ x^5*(1 - 6*A^5 + 20*A^12 - 50*A^21 + 105*A^32 - 196*A^45 +...)
+ x^6*(1 - 7*A^6 + 27*A^14 - 182*A^24 + 378*A^36 - 714*A^50 +...)
+...
where coefficient of x^n * A^(k*(n+k-1)) = A293600(n,k) = [z^k] (1-z)/(1+z)^n.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..600 (terms 1..280 from Vaclav Kotesovec)

Crossrefs

Cf. A293600.

Programs

PARI

{a(n) = my(A=x-x^2,P=-1,Q=1,Ox=x*O(x^n)); for(i=0,n,
P = sum(m=0,n+1, (x - A^m +Ox)^(m+1) );
Q = sum(m=1,sqrtint(n+9), (-1)^(m-1) * A^(m*(m-1)) / (1 - x*A^m +Ox)^(m-1) );
A = P + Q;); polcoeff(A,n)}
for(n=1,40,print1(a(n),", "))

PARI

{ A293600(n, k) = my(z=x+x*O(x^k)); polcoeff( (1-z)/(1+z)^n, k) }
{ a(n) = my(A=x-x^2 +x*O(x^n)); for(i=0,n,
A = sum(m=1,n, x^m * sum(k=0,n\m+1, A293600(m,k) * A^(k*(m+k-1)) )););
polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

(1) A(x) = Sum_{-oo..+oo} (x - A(x)^n)^(n+1).

(2) A(x) = x * Sum_{-oo..+oo} (x - A(x)^n)^n.

(3) A(x) = x/(1-x) + Sum_{n>=1} (-1)^n * x * A(x)^(n^2) * (2 - x*A(x)^n)/(1 - x*A(x)^n)^(n+1).

(4) A(x) = Sum_{n>=1} x^n * Sum_{k>=0} A293600(n,k) * A(x)^(k*(n+k-1)), where A293600(n,k) = [z^k] (1-z)/(1+z)^n.

(5) A(x) = P(x) + Q(x), where

P(x) = Sum_{n>=0} (x - A(x)^n)^(n+1),

Q(x) = Sum_{n>=1} (-1)^(n-1) * A(x)^(n*(n-1)) / (1 - x*A(x)^n)^(n-1).

a(n) ~ -(-1)^n * c * d^n / n^(3/2), where d = 4.2403058670395262369476988786... and c = 0.19755290815424088971... - Vaclav Kotesovec, Oct 20 2017