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 A216814 Given n and a constant C, define a sequence b(m) by the recurrence in the comments; a(n) = smallest positive integer C such that for some prime p the denominators of all b(m) are powers of p (conjectured). 0
 2, 4, 10, 12, 84, 264, 990, 2860, 9724, 18564, 117572, 45220, 19380, 1782960, 6463230, 25092540, 58549260, 95527740 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS The sequence b(m) is defined by b(1)=C, and for m>=2, b(m) = (1/(2*binomial(m+n-1,m-1))) * Sum_{k=1..m-1} binomial(m+n-1,m-k-1)*binomial(m+n-1,k-1)*b(k)*b(m-k). For n=2..19, the corresponding primes p are 3, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 23, 23, 29, 31, 31, 31, 37. The terms up to a(19) have been obtained by generating the first 2000 terms of the relative sequences. - Giovanni Resta, Oct 07 2019 LINKS Tewodros Amdeberhan, Victor H. Moll and Christophe Vignat, A probabilistic interpretation of a sequence related to Narayana Polynomials, arXiv:1202.1203 [math.NT], 2012. Tewodros Amdeberhan, Victor H. Moll and Christophe Vignat, A probabilistic interpretation of a sequence related to Narayana Polynomials, Online Journal of Analytic Combinatorics, Issue 8, 2013. - From N. J. A. Sloane, Sep 19 2012 MATHEMATICA ppQ[n_] := n == 1 || PrimePowerQ[n]; isOk[n_, c_, mmax_] := Block[{d, p=1, ret=True, vb = 0 Range@ mmax}, vb[] = c; Do[ vb[[m]] = (1/(2 * Binomial[m+n-1, m-1]) Sum[ Binomial[ m+n-1, m-k-1] * Binomial[m+n-1, k-1] * vb[[k]]*vb[[m - k]], {k, m-1}]); If[! ppQ[d = Denominator[vb[[m]]]], ret = False; Break[]]; If[d != 1, d = FactorInteger[d][[1, 1]]; If[p == 1, p = d, If[p != d, ret = False; Break[]]]], {m, 2, mmax}]; ret]; a[n_] := Block[{c = 1}, While[! isOk[n, c, 100], c++]; c]; a/@ Range[2, 10] (* Giovanni Resta, Oct 07 2019 *) PROG (PARI) ispp(n) = (n==1) || isprimepower(n); isokC(n, C, mmax) = {my(vb = vector(mmax)); vb = C; for (m=2, mmax, vb[m] = (1/(2*binomial(m+n-1, m-1))*sum(k=1, m-1, binomial(m+n-1, m-k-1)*binomial(m+n-1, k-1)*vb[k]*vb[m-k])); if (!ispp(denominator(vb[m])), return (0)); ); return (1); } a(n) = {my(C=1, mmax = 1000); while(!isokC(n, C, mmax), C++); C; } \\ Michel Marcus, Sep 29 2019 CROSSREFS Sequence in context: A301338 A181495 A092367 * A180427 A127591 A100912 Adjacent sequences:  A216811 A216812 A216813 * A216815 A216816 A216817 KEYWORD nonn,more AUTHOR N. J. A. Sloane, Sep 20 2012 EXTENSIONS Conjectured added to name and a(10)-a(15) from Michel Marcus, Oct 06 2019 a(16)-a(19) from Giovanni Resta, Oct 07 2019 STATUS approved

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Last modified September 24 21:05 EDT 2021. Contains 347651 sequences. (Running on oeis4.)