A268137 -id:A268137 - 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}]

A268138 a(n) = (Sum_{k=0..n-1} A001850(k)*A001003(k+1))/n.

Original entry on oeis.org

1, 5, 51, 747, 13245, 264329, 5721415, 131425079, 3159389817, 78729848397, 2019910325499, 53087981674275, 1423867359013749, 38855956977763857, 1076297858301372687, 30203970496501504239, 857377825323716359665, 24586286492003180067989, 711463902659879056604995, 20756358426519694831851227

Offset: 1

Zhi-Wei Sun, Jan 26 2016

nonn

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} binomial(n,k)*binomial(n+k,k)*x^k = Sum_{k=0..n} binomial(n,k)^2*x^k*(x+1)^(n-k) for n >= 0, and s_n(x) = Sum_{k=1..n} (binomial(n,k)*binomial(n,k-1)/n)*x^(k-1)*(x+1)^(n-k) = (Sum_{k=0..n} binomial(n,k)*binomial(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

			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.

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.

Cf. A001003, A001850, A006318, A268136, A268137, A182626, A358388.

A001850 := n -> LegendreP(n, 3); seq(((3*(2*n+1)*A001850(n)*A001850(n-1)-n*A001850(n-1)^2)/(n+1) - A001850(n)^2)/4, n=1..20); # Mark van Hoeij, Nov 12 2022
# Alternative (which also gives an integer for n = 0):
f := n -> hypergeom([-n, -n], [1], 2):          # A001850
h := n -> hypergeom([-n,  n], [1], 2):          # A182626
g := n -> hypergeom([-n,  n, 1/2], [1, 1], -8): # A358388
a := n -> (f(n)*((3*n + 1)*f(n) - (-1)^n*(6*n + 3)*h(n)) - n*g(n))/(2*n + 2):
seq(simplify(a(n)), n = 1..20); # Peter Luschny, Nov 13 2022

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}]

a(n) = ((3*(2*n+1)*A001850(n)*A001850(n-1) - n*A001850(n-1)^2)/(n+1) - A001850(n)^2)/4. - Mark van Hoeij, Nov 12 2022

G.f.: (1-(1+1/x)*Int((1-34*x+x^2)^(1/2) * hypergeom([-1/2,1/2],[1], -32*x/(1-34*x+x^2))/((1-x)*(1+x)^2),x))/4. - Mark van Hoeij, Nov 28 2024

cp's OEIS Frontend

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

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