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%I #29 Oct 07 2019 11:04:12
%S 2,4,10,12,84,264,990,2860,9724,18564,117572,45220,19380,1782960,
%T 6463230,25092540,58549260,95527740
%N 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).
%C 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).
%C 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.
%C The terms up to a(19) have been obtained by generating the first 2000 terms of the relative sequences. - _Giovanni Resta_, Oct 07 2019
%H Tewodros Amdeberhan, Victor H. Moll and Christophe Vignat, <a href="https://arxiv.org/abs/1202.1203">A probabilistic interpretation of a sequence related to Narayana Polynomials</a>, arXiv:1202.1203 [math.NT], 2012.
%H Tewodros Amdeberhan, Victor H. Moll and Christophe Vignat, <a href="https://web.math.rochester.edu/misc/ojac/vol8/68.pdf">A probabilistic interpretation of a sequence related to Narayana Polynomials</a>, Online Journal of Analytic Combinatorics, Issue 8, 2013. - From _N. J. A. Sloane_, Sep 19 2012
%t ppQ[n_] := n == 1 || PrimePowerQ[n]; isOk[n_, c_, mmax_] := Block[{d, p=1, ret=True, vb = 0 Range@ mmax}, vb[[1]] = 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 *)
%o (PARI) ispp(n) = (n==1) || isprimepower(n);
%o isokC(n, C, mmax) = {my(vb = vector(mmax)); vb[1] = 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);}
%o a(n) = {my(C=1, mmax = 1000); while(!isokC(n, C, mmax), C++); C;} \\ _Michel Marcus_, Sep 29 2019
%K nonn,more
%O 2,1
%A _N. J. A. Sloane_, Sep 20 2012
%E Conjectured added to name and a(10)-a(15) from _Michel Marcus_, Oct 06 2019
%E a(16)-a(19) from _Giovanni Resta_, Oct 07 2019
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