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[[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 *)
PROG
(PARI) ispp(n) = (n==1) || isprimepower(n);
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); }
a(n) = {my(C=1, mmax = 1000); while(!isokC(n, C, mmax), C++); C; } \\ Michel Marcus, Sep 29 2019
CROSSREFS
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