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A292088 G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} x^n * A(x)^n * (A(x)^n + x^n)^n. 2
1, 0, 4, 16, 90, 552, 3818, 27256, 200812, 1515912, 11695500, 91752936, 729850840, 5873414168, 47739736006, 391396504592, 3233190109306, 26886605900496, 224911668468892, 1891436025156672, 15982645615936990, 135640664826165488, 1155711422893896044, 9882880183890197352, 84794414367362654416, 729778564590247004856, 6298814833598391313930 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare g.f. to: Sum_{n=-oo..+oo} x^n * (c - x^n)^n = 0 for fixed |c| > 0.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..100

FORMULA

G.f. A = A(x) = Sum_{n>=0} a(n)*x^n satisfies:

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

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

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

EXAMPLE

G.f.: A(x) = 1 + 4*x^2 + 16*x^3 + 90*x^4 + 552*x^5 + 3818*x^6 + 27256*x^7 + 200812*x^8 + 1515912*x^9 + 11695500*x^10 + 91752936*x^11 + 729850840*x^12 + 5873414168*x^13 + 47739736006*x^14 + 391396504592*x^15 + 3233190109306*x^16 +...

Given A = A(x), define

P(x) = (x*A)*(A + x) + (x*A)^2*(A^2 + x^2)^2 + (x*A)^3*(A^3 + x^3)^3 + (x*A)^4*(A^4 + x^4)^4 + (x*A)^5*(A^5 + x^5)^5 +...+ (x*A)^n*(A^n + x^n)^n +...

Q(x) = 1/(A + x) + (x*A)^2/(A^2 + x^2)^2 + (x*A)^6/(A^3 + x^3)^3 + (x*A)^12/(A^4 + x^4)^4 + (x*A)^20/(A^5 + x^5)^5 +...+ (x*A)^(n^2-n)/(A^n + x^n)^n +...

then Q(x) = 1 - P(x).

Explicitly,

P(x) = x + 2*x^2 + 9*x^3 + 63*x^4 + 357*x^5 + 2411*x^6 + 17101*x^7 + 126047*x^8 + 950172*x^9 + 7324084*x^10 + 57423493*x^11 + 456717652*x^12 + 3675758545*x^13 + 29884252434*x^14 + 245091410895*x^15 + 2025466163355*x^16 + 16851425417853*x^17 + 141038441106711*x^18 + 1186738922293689*x^19 + 10033676061349606*x^20 +...

Q(x) = 1 - x - 2*x^2 - 9*x^3 - 63*x^4 - 357*x^5 - 2411*x^6 +...

PROG

(PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0); A[#A] = Vec( sum(m=-#A-1, #A+1, x^m*Ser(A)^m * (Ser(A)^m + x^m)^m))[#A]); A[n+1] }

for(n=0, 40, print1(a(n), ", "))

CROSSREFS

Sequence in context: A235166 A013030 A124962 * A221818 A331648 A009568

Adjacent sequences:  A292085 A292086 A292087 * A292089 A292090 A292091

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 08 2017

STATUS

approved

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Last modified May 7 06:57 EDT 2021. Contains 343636 sequences. (Running on oeis4.)