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A326562 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, 6, 26, 134, 775, 4856, 32359, 226688, 1656745, 12566075, 98550684, 797062686, 6635504831, 56781439908, 498946126131, 4498769217830, 41598980113524, 394300130478239, 3829670219184141, 38100703672385734, 388140101200555331, 4047307115413115559, 43180809390468971803, 471163737793390252840, 5255377025199543952036, 59891933515705554763680, 697006510462415153074548 (list; graph; refs; listen; history; text; internal format)
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 + 6*x^3 + 26*x^4 + 134*x^5 + 775*x^6 + 4856*x^7 + 32359*x^8 + 226688*x^9 + 1656745*x^10 + 12566075*x^11 + 98550684*x^12 + ...
such that the following sums are equal
B(x) = A(x) + A(x)*x + A(x)^3*x^2 + A(x)^7*x^3 + A(x)^13*x^4 + A(x)^21*x^5 + A(x)^31*x^6 + A(x)^43*x^7 + A(x)^57*x^8 + ... + A(x)^(n*(n-1)+1)*x^n + ...
and
B(x) = 1 + 2*x + (1 + A(x))^2*x^2 + (1 + A(x)^2)^3*x^3 + (1 + A(x)^3)^4*x^4 + (1 + A(x)^4)^5*x^5 + (1 + A(x)^5)^6*x^6 + ... + (1 + A(x)^(n+1))^n*x^n + ...
also
B(x) = 1/(1 - x) + x/(1 - x*A(x))^2 + A(x)^2*x^2/(1 - x*A(x)^2)^3 + A(x)^6*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 + 4*x^2 + 12*x^3 + 49*x^4 + 240*x^5 + 1328*x^6 + 8014*x^7 + 51691*x^8 + 351839*x^9 + 2505762*x^10 + 18563322*x^11 + 142460948*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
Sequence in context: A030876 A159667 A370215 * A030957 A030898 A002788
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 23 2019
STATUS
approved

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)