cp's OEIS Frontend

A268137 a(n) = (1/n)*Sum_{k=0..n-1} A001850(k)*A245769(k).

%I A268137 #26 Nov 03 2019 16:29:15
%S A268137 -1,1,31,417,5919,97217,1828479,38085249,853450367,20174707521,
%T A268137 496690317855,12626836592289,329476040177439,8785359461936769,
%U A268137 238587766484265471,6581966817521388033,184067922884292651519,5209333642085984431489,148992465188631205367071,4301514890878664802287777
%N A268137 a(n) = (1/n)*Sum_{k=0..n-1} A001850(k)*A245769(k).
%C A268137 Conjecture: (i) All the terms are odd integers. For any prime p, if p == 3 (mod 4) then a(p) == -5 (mod p^2), otherwise a(p) == -1 (mod p).
%C A268137 (ii) For n = 0,1,2,... let D_n(x) = Sum_{k=0..n} binomial(n,k)*binomial(n+k,k)*x^k and R_n(x) = Sum_{k=0..n} binomial(n,k)*binomial(n+k,k)*x^k/(2k-1). For any positive integer n, all the coefficients of the polynomial (1/n)*Sum_{k=0..n-1} D_k(x)*R_k(x) are integral and the polynomial is irreducible over the field of rational numbers.
%H A268137 Zhi-Wei Sun, <a href="/A268137/b268137.txt">Table of n, a(n) for n = 1..100</a>
%H A268137 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1408.5381">Two new kinds of numbers and related divisibility results</a>, preprint, arXiv:1408.5381 [math.NT], 2014.
%H A268137 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1602.00574">Arithmetic properties of Delannoy numbers and Schröder numbers</a>, preprint, arXiv:1602.00574 [math.CO], 2016.
%e A268137 a(3) = 31 since (A001850(0)*A245769(0) + A001850(1)*A245769(1) + A001850(2)*A245769(2))/3 = (1*(-1) + 3*1 + 13*7)/3 = 93/3 = 31.
%t A268137 d[n_]:=d[n]=Sum[Binomial[n,k]Binomial[n+k,k],{k,0,n}]
%t A268137 R[n_]:=R[n]=Sum[Binomial[n,k]Binomial[n+k,k]/(2k-1),{k,0,n}]
%t A268137 a[n_]:=a[n]=Sum[d[k]*R[k],{k,0,n-1}]/n
%t A268137 Do[Print[n," ",a[n]],{n,1,20}]
%Y A268137 Cf. A001850, A245769, A268136, A268138.
%K A268137 sign
%O A268137 1,3
%A A268137 _Zhi-Wei Sun_, Jan 26 2016