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URL: https://oeis.org/A326561

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A326561
G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n*(n+1)+1) * x^n = Sum_{n>=0} (A(x)^(n+1) + 1)^n * x^n.
7
1, 1, 2, 8, 43, 268, 1831, 13354, 102352, 816241, 6728037, 57056328, 496185294, 4414563737, 40114493041, 371845437684, 3513262944971, 33815076029376, 331454445914861, 3308183361941640, 33620505978224843, 347942136527114740, 3667458554727506170, 39378668472879902163, 430803711668467138362, 4802830669726993050928, 54572510428547279296599
OFFSET
0,3
LINKS
FORMULA
G.f. A(x) allows the following sums to be equal:
(1) B(x) = Sum_{n>=0} A(x)^(n*(n+1)+1) * x^n.
(2) B(x) = Sum_{n>=0} (A(x)^(n+1) + 1)^n * x^n.
(3) B(x) = Sum_{n>=0} A(x)^(n*(n+1)) * x^n / (1 - x*A(x)^n)^(n+1).
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 43*x^4 + 268*x^5 + 1831*x^6 + 13354*x^7 + 102352*x^8 + 816241*x^9 + 6728037*x^10 + 57056328*x^11 + 496185294*x^12 + ...
such that the following sums are equal
B(x) = A(x) + A(x)^3*x + A(x)^7*x^2 + A(x)^13*x^3 + A(x)^21*x^4 + A(x)^31*x^5 + A(x)^43*x^6 + A(x)^57*x^7 + A(x)^73*x^8 + ... + A(x)^(n*(n+1)+1)*x^n + ...
and
B(x) = 1 + (1 + A(x)^2)*x + (1 + A(x)^3)^2*x^2 + (1 + A(x)^4)^3*x^3 + (1 + A(x)^5)^4*x^4 + (1 + A(x)^6)^5*x^5 + ... + (1 + A(x)^(n+1))^n*x^n + ...
also
B(x) = 1/(1 - x) + A(x)^2*x/(1 - x*A(x))^2 + A(x)^6*x^2/(1 - x*A(x)^2)^3 + A(x)^12*x^3/(1 - x*A(x)^3)^4 + ... + A(x)^(n*(n+1))*x^n/(1 - x*A(x)^n)^(n+1) + ...
where
B(x) = 1 + 2*x + 6*x^2 + 25*x^3 + 129*x^4 + 764*x^5 + 4977*x^6 + 34770*x^7 + 256358*x^8 + 1973671*x^9 + 15750935*x^10 + 129624972*x^11 + 1095963211*x^12 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, (Ser(A)^(m+1) + 1)^m*x^m - Ser(A)^(m^2+m+1)*x^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 12 2019
STATUS
approved