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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

Table of n, a(n) for n=2..19.

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

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 November 21 22:40 EST 2019. Contains 329383 sequences. (Running on oeis4.)