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A318644
G.f. A(x) satisfies: A(x) = Sum_{n>=0} (1+x)^(n*(n+1)/2) * x^n / A(x)^n.
3
1, 1, 1, 1, 2, 4, 11, 32, 106, 376, 1433, 5782, 24574, 109393, 508026, 2453256, 12285347, 63656731, 340626704, 1879183856, 10672897341, 62323897482, 373748877678, 2299318074357, 14497472040378, 93599428822052, 618278575554155, 4175348680420942, 28806364292660618, 202899326988089615, 1458130019936912105, 10685096640964659318
OFFSET
0,5
LINKS
FORMULA
G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n * (1+x)^(n*(n+1)/2) / A(x)^n.
(2) 1 + x = Sum_{n>=0} x^n * (1+x)^(n*(n-1)/2) / A(x)^n.
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 11*x^6 + 32*x^7 + 106*x^8 + 376*x^9 + 1433*x^10 + 5782*x^11 + 24574*x^12 + 109393*x^13 + 508026*x^14 + ...
such that
A(x) = 1 + (1+x)*x/A(x) + (1+x)^3*x^2/A(x)^2 + (1+x)^6*x^3/A(x)^3 + (1+x)^10*x^4/A(x)^4 + (1+x)^15*x^5/A(x)^5 + (1+x)^21*x^6/A(x)^6 + (1+x)^28*x^7/A(x)^7 + ... + (1+x)^(n*(n+1)/2) * x^n / A(x)^n + ...
Also
1 + x = 1 + x/A(x) + (1+x)*x^2/A(x)^2 + (1+x)^3*x^3/A(x)^3 + (1+x)^6*x^4/A(x)^4 + (1+x)^10*x^5/A(x)^5 + (1+x)^15*x^6/A(x)^6 + (1+x)^21*x^7/A(x)^7 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec(sum(n=0, #A, (1+x +x*O(x^#A))^(n*(n+1)/2) * x^n/Ser(A)^n ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Sequence in context: A268322 A375641 A148171 * A113774 A124504 A056324
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 07 2018
STATUS
approved