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A268138 a(n) = (Sum(A001850(k)*A001003(k+1), k = 0..n-1))/n 3
1, 5, 51, 747, 13245, 264329, 5721415, 131425079, 3159389817, 78729848397, 2019910325499, 53087981674275, 1423867359013749, 38855956977763857, 1076297858301372687, 30203970496501504239, 857377825323716359665, 24586286492003180067989, 711463902659879056604995, 20756358426519694831851227 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: (i) All the terms are odd integers. Also, p | a(p) for any odd prime p.

(ii) Let D_n(x) = Sum_{k=0..n}binom(n,k)binom(n+k,k)*x^k = Sum_{k=0..n}binom(n,k)^2*x^k*(x+1)^(n-k) for n >= 0, and s_n(x) = Sum_{k=1..n}binom(n,k)binom(n,k-1)/n*x^(k-1)*(x+1)^(n-k) = (Sum_{k=0..n}binom(n,k)binom(n+k,k)x^k/(k+1))/(x+1) for n > 0. Then, for any positive integer n, all the coefficients of the polynomial 1/n*Sum_{k=0..n-1}D_k(x)*s_{k+1}(x) are integral and the polynomial is irreducible over the field of rational numbers.

The conjecture was essentially proved by the author in arXiv:1602.00574, except for the irreducibility of (Sum_{k=0..n-1}D_k(x)*s_{k+1}(x))/n. - Zhi-Wei Sun, Feb 01 2016

LINKS

Zhi-Wei Sun, Table of n, a(n)for n = 1..100

Zhi-Wei Sun, On Delannoy numbers and Schroder numbers, J. Number Theory 131(2011), no.12, 2387-2397.

Zhi-Wei Sun, Arithmetic properties of Delannoy numbers and Schröder numbers, preprint, arXiv:1602.00574 [math.CO], 2016.

EXAMPLE

a(3) = 51 since (A001850(0)*A001003(1) + A001850(1)*A001003(2) + A001850(2)*A001003(3))/3 = (1*1 + 3*3 + 13*11)/3 = 153/3 = 51.

MATHEMATICA

d[n_]:=Sum[Binomial[n, k]Binomial[n+k, k], {k, 0, n}]

s[n_]:=Sum[Binomial[n, k]Binomial[n, k-1]/n*2^(k-1), {k, 1, n}]

a[n_]:=Sum[d[k]s[k+1], {k, 0, n-1}]/n

Table[a[n], {n, 1, 20}]

CROSSREFS

Cf. A001003, A001850, A006318, A268136, A268137.

Sequence in context: A245926 A190734 A154886 * A145162 A187235 A318192

Adjacent sequences:  A268135 A268136 A268137 * A268139 A268140 A268141

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 26 2016

STATUS

approved

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Last modified October 22 12:30 EDT 2019. Contains 328318 sequences. (Running on oeis4.)