A268138 -id:A268138 - cp's OEIS frontend

A268136 a(n) = (3/n)*Sum_{k=0..n-1} A245769(k)^2.

3, 3, 51, 507, 4947, 58243, 841443, 14240763, 269512483, 5524472451, 120183938835, 2738420763131, 64760819179635, 1579226738429187, 39515677808716739, 1010750709382934523, 26349289260686093379, 698387854199468231427, 18783213754115549685747, 511772677524431483886075

Offset: 1

Zhi-Wei Sun, Jan 26 2016

nonn

Conjecture: (i) All the terms are odd integers.

(ii) For n = 0,1,2,... let R_n(x) denote the polynomial sum_{k=0..n} binomial(n,k)*binomial(n+k,k)*x^k/(2k-1). Then, for each n = 1,2,3,.., all the coefficients of the polynomial (3/n)*Sum_{k=0..n-1} R_k(x)^2 are integral and the polynomial is irreducible over the field of rational numbers.

			a(3) = 51 since (3/3)*(A245769(0)^2 + A245769(1)^2 + A245769(2)^2) = (-1)^2 + 1^2 + 7^2 = 51.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100
Zhi-Wei Sun, Two new kinds of numbers and related divisibility results, preprint, arXiv:1408.5381 [math.NT], 2014.

Cf. A000290, A245769, A268137, A268138.

R[n_]:=Sum[Binomial[n,k]Binomial[n+k,k]/(2k-1),{k,0,n}]
a[n_]:=Sum[R[k]^2,{k,0,n-1}]*3/n
Do[Print[n," ",a[n]],{n,1,20}]

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

Original entry on oeis.org

-1, 1, 31, 417, 5919, 97217, 1828479, 38085249, 853450367, 20174707521, 496690317855, 12626836592289, 329476040177439, 8785359461936769, 238587766484265471, 6581966817521388033, 184067922884292651519, 5209333642085984431489, 148992465188631205367071, 4301514890878664802287777

Offset: 1

Zhi-Wei Sun, Jan 26 2016

sign

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).

(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.

			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.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100
Zhi-Wei Sun, Two new kinds of numbers and related divisibility results, preprint, arXiv:1408.5381 [math.NT], 2014.
Zhi-Wei Sun, Arithmetic properties of Delannoy numbers and Schröder numbers, preprint, arXiv:1602.00574 [math.CO], 2016.

Cf. A001850, A245769, A268136, A268138.

d[n_]:=d[n]=Sum[Binomial[n,k]Binomial[n+k,k],{k,0,n}]
R[n_]:=R[n]=Sum[Binomial[n,k]Binomial[n+k,k]/(2k-1),{k,0,n}]
a[n_]:=a[n]=Sum[d[k]*R[k],{k,0,n-1}]/n
Do[Print[n," ",a[n]],{n,1,20}]

cp's OEIS Frontend

A268136 a(n) = (3/n)*Sum_{k=0..n-1} A245769(k)^2.

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

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cp's OEIS Frontend

A268136 a(n) = (3/n)*Sum_{k=0..n-1} A245769(k)^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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