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 A237646 G.f.: exp( Sum_{n>=1} A163659(n^3)*x^n/n ), where x*exp(Sum_{n>=1} A163659(n)*x^n/n) = S(x) is the g.f. of Stern's diatomic series (A002487). 4
 1, 1, 8, 7, 63, 56, 329, 273, 1736, 1463, 7511, 6048, 32585, 26537, 124440, 97903, 475287, 377384, 1658881, 1281497, 5783960, 4502463, 18825023, 14322560, 61171649, 46849089, 188181672, 141332583, 577889023, 436556440, 1696298665, 1259742225, 4970284200, 3710541975, 14019036535, 10308494560 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Compare to the g.f. of A195586. LINKS FORMULA G.f.: exp( Sum_{n>=1} A237649(n)*x^n/n ), where A237649(n) = A163659(n^3). G.f. A(x) satisfies: (1) A(x) = (1+x+x^2) * (1+x^2+x^4)^3 * A(x^2)^4. (2) A(x) = (1+x+x^2) * Product_{n>=0} ( 1 + x^(2*2^n) + x^(4*2^n) )^(7*4^n). (3) A(x) / A(-x) = (1+x+x^2) / (1-x+x^2). Bisections: let A(x) = B(x^2) + x*C(x^2), then (4) B(x) = (1+x) * C(x). (5) C(x) = (1+x+x^2)^7 * C(x^2)^4. (6) A(x) = (1+x+x^2) * C(x^2). (7) A(x)^7 = C(x) * C(x^2)^3. (8) A(x)^4 = C(x) / (1+x+x^2)^3. (9) A(x)^3 = ( C(x)/A(x) - C(x^2)^4/A(x^2)^4 ) / (6*x + 14*x^3 + 6*x^5). EXAMPLE G.f.: A(x) = 1 + x + 8*x^2 + 7*x^3 + 63*x^4 + 56*x^5 + 329*x^6 + 273*x^7 +... where log(A(x)) = x + 15*x^2/2 - 2*x^3/3 + 127*x^4/4 + x^5/5 - 30*x^6/6 + x^7/7 + 1023*x^8/8 +...+ A237649(n)*x^n/n +... Bisections: let A(x) = B(x^2) + x*C(x^2), then: B(x) = 1 + 8*x + 63*x^2 + 329*x^3 + 1736*x^4 + 7511*x^5 + 32585*x^6 +... C(x) = 1 + 7*x + 56*x^2 + 273*x^3 + 1463*x^4 + 6048*x^5 + 26537*x^6 + 97903*x^7 + 377384*x^8 + 1281497*x^9 + 4502463*x^10 +...+ A237647(n)*x^n +... Note that C(x)^(1/7) = (1+x+x^2) * C(x^2)^(4/7) is an integer series: C(x)^(1/7) = 1 + x + 5*x^2 + 4*x^3 + 30*x^4 + 26*x^5 + 106*x^6 + 80*x^7 + 459*x^8 + 379*x^9 + 1451*x^10 + 1072*x^11 + 5210*x^12 +...+ A237648(n)*x^n +... Also, C(x) / (1+x+x^2)^3 = A(x)^4: A(x)^4 = 1 + 4*x + 38*x^2 + 128*x^3 + 817*x^4 + 2536*x^5 + 12890*x^6 +... Further, C(x)*C(x^2)^3 = A(x)^7: A(x)^7 = 1 + 7*x + 77*x^2 + 420*x^3 + 2954*x^4 + 13986*x^5 + 78414*x^6 +... The g.f. may be expressed by the product: A(x) = (1+x+x^2) * (1+x^2+x^4)^7 * (1+x^4+x^8)^28 * (1+x^8+x^16)^112 * (1+x^16+x^32)^448 *...* (1 + x^(2*2^n) + x^(4*2^n))^(7*4^n) *... PROG (PARI) {A163659(n)=if(n<1, 0, if(n%3, 1, -2)*sigma(2^valuation(n, 2)))} {a(n)=polcoeff(exp(sum(k=1, n, A163659(k^3)*x^k/k)+x*O(x^n)), n)} for(n=0, 40, print1(a(n), ", ")) CROSSREFS Cf. A237647, A237648, A237649, A195586. Sequence in context: A138809 A286460 A317231 * A288188 A038285 A261117 Adjacent sequences:  A237643 A237644 A237645 * A237647 A237648 A237649 KEYWORD nonn AUTHOR Paul D. Hanna, May 03 2014 STATUS approved

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Last modified November 27 03:56 EST 2021. Contains 349345 sequences. (Running on oeis4.)