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 A245658 G.f. A(x) satisfies: A(x)^3 = Sum_{n>=0} A247031(n)*x^n. 2
 1, 1, 22, 1624, 264962, 79136637, 38671111558, 28642761340956, 30413158977739302, 44371247722932580948, 86030010617294195269924, 215785986252313362542154058, 684854225048414942925120331598, 2699657503162253569920254747627596, 13010506207186236974375590663943378970 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The terms of A247031 satisfy: 1 = Sum_{n>=0} A247031(n)*x^n * [Sum_{k=0..n+1} C(n+1, k)^3 * (-x)^k]^3. Limit a(n)/A247031(n) = 1/3. a(n) == 1 (mod 2) iff n = (4^k - 1)/3 for k>=0 (conjecture). a(n) == 2 (mod 3) iff A247031(3*n) == 2 for n>=0 (mod 3) (conjecture), where A247031 is the self-convolution cube of this sequence. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..200 EXAMPLE G.f.: A(x) = 1 + x + 22*x^2 + 1624*x^3 + 264962*x^4 + 79136637*x^5 +... where the coefficients of the cube of the g.f. A(x)^3 = 1 + 3*x + 69*x^2 + 5005*x^3 + 806148*x^4 + 239220375*x^5 +... satisfies: 1 = 1*(1-x)^3 + 3*x*(1-2^3*x+x^2)^3 + 69*x^2*(1-3^3*x+3^3*x^2-x^3)^3 + 5005*x^3*(1-4^3*x+6^3*x^2-4^3*x^3+x^4)^3 + 806148*x^4*(1-5^3*x+10^3*x^2-10^3*x^3+5^3*x^4-x^5)^3 +... OBSERVATIONS. The terms of this sequence are odd at positions: [0, 1, 5, 21, 85, 341, ..., (4^k-1)/3, ...]. The terms of this sequence are congruent to 2 modulo 3 at positions: [4,12,31,36,37,38,40,58,85,93,108,109,110,111,114,120,166,174,247,255,...]. The terms of A247031 are congruent to 2 modulo 3 at positions: [12,36,93,108,111,114,120,174,255,279,324,327,330,333,342,360,...]. PROG (PARI) {A247031(n)=if(n==0, 1, -polcoeff(sum(m=0, n-1, A247031(m)*x^m*sum(k=0, m+1, binomial(m+1, k)^3 * (-x)^k +x*O(x^n) )^3 ), n))} {a(n)=polcoeff(sum(m=0, n, A247031(m)*x^m +x*O(x^n))^(1/3), n)} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A247031. Sequence in context: A033526 A078399 A243478 * A277664 A199836 A196705 Adjacent sequences: A245655 A245656 A245657 * A245659 A245660 A245661 KEYWORD nonn AUTHOR Paul D. Hanna, Sep 13 2014 STATUS approved

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Last modified September 16 23:59 EDT 2024. Contains 375984 sequences. (Running on oeis4.)