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A206740
G.f.: 1/(1 - x/(1 - x^3/(1 - x^6/(1 - x^10/(1 - x^15/(1 - x^21/(1 -...- x^(n*(n+1)/2)/(1 -...))))))), a continued fraction.
4
1, 1, 1, 1, 2, 3, 4, 6, 9, 13, 20, 30, 44, 66, 99, 147, 220, 329, 490, 732, 1095, 1634, 2440, 3646, 5444, 8130, 12146, 18139, 27089, 40463, 60434, 90258, 134811, 201349, 300721, 449153, 670844, 1001939, 1496467, 2235080, 3338227, 4985868, 7446739, 11122179
OFFSET
0,5
FORMULA
G.f.: 1/Q(0) , where Q(k) = 1 - x^((2*k+1)*(2*k+2)/2)/(1 - x^((2*k+2)*(2*k+3)/2)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Sep 10 2013
a(n) ~ c * d^n, where d = 1.49356638691558702616975760297981328... and c = 0.35853801643147450974166770910994348... - Vaclav Kotesovec, Aug 24 2017
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 6*x^7 + 9*x^8 +...
MATHEMATICA
nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[nmax + 1]*(Range[nmax + 1] + 1)/2)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 24 2017 *)
PROG
(PARI) {a(n)=local(CF=1+x*O(x^n), M=sqrtint(n+1)); for(k=0, M, CF=1/(1-x^((M-k+1)*(M-k+2)/2)*CF)); polcoeff(CF, n, x)}
for(n=0, 55, print1(a(n), ", "))
CROSSREFS
Cf. A206739.
Sequence in context: A136423 A215245 A078932 * A172161 A117791 A022860
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 12 2012
STATUS
approved