A358388 -id:A358388 - cp's OEIS frontend

A178808 a(n) = (1/n^2) * Sum_{k = 0..n-1} (2k+1)(D_k)^2, where D_0, D_1, ... are central Delannoy numbers given by A001850.

1, 7, 97, 1791, 38241, 892039, 22092673, 571387903, 15271248769, 418796912007, 11725812711009, 333962374092543, 9648543623050593, 282164539499639559, 8338391167566634497, 248661515283002490879, 7474768663941435203073

Offset: 1

Zhi-Wei Sun, Jun 16 2010

nonn

On Jun 14 2010, Zhi-Wei Sun conjectured that a(n) = (1/n^2) * Sum_{k = 0..n-1} (2*k+1)*(D_k)^2 is always an integer and that p^2*a(p) = p^2 - 4*p^3*q_p(2) - 2*p^4*q_p(2)^2 (mod p^5) for any prime p > 3, where q_p(2) denotes the Fermat quotient (2^(p-1) - 1)/p (see Sun, Remark 4.3, p. 26, 2014). He also conjectured that Sum_{k = 0..n-1} (2*k+1)*(-1)^k*(D_k)^2 == 0 (mod n*D_n/(3,D_n)) for all n = 1,2,3,....

The fact that a(n) is an integer follows directly from the formulas for a(n) in the formula section below. - Mark van Hoeij, Nov 13 2022

			For n = 3 we have a(3) = (D_0^2 + 3*D_1^2 + 5*D_2^2)/3^2 = (1 + 3*3^2 + 5*13^2)/3^2 = 97.

G. C. Greubel, Table of n, a(n) for n = 1..500
Zhi-Wei Sun, Arithmetic properties of Apery numbers and central Delannoy numbers, arXiv:1006.2776 [math.NT], 2011.
Zhi-Wei Sun, Congruences involving generalized central trinomial coefficients, Sci. China Math. 57 (2014), no. 7, 1375-1400; arXiv:1008.3887 [math.NT], 2010-2014.

Cf. A001850, A178790, A178791, A173774, A358388.

A001850 := n -> LegendreP(n, 3); seq((6*A001850(n)*A001850(n-1)-A001850(n)^2-A001850(n-1)^2)/8, n=1..20); # Mark van Hoeij, Nov 12 2022
# Alternative:
g := n -> hypergeom([n, -n, 1/2], [1, 1], -8): # A358388
f := n -> hypergeom([-n, -n], [1], 2):         # A001850
a := n -> (3*f(n)*f(n-1) - g(n)) / 4:
seq(simplify(a(n)), n = 1..17); # Peter Luschny, Nov 13 2022

DD[n_]:=Sum[Binomial[n+k,2k]Binomial[2k,k],{k,0,n}]; SS[n_]:= Sum[(2k+1)*DD[k]^2,{k,0,n-1}]/n^2; Table[SS[n],{n,1,25}]
Table[Sum[(2k+1)*JacobiP[k,0,0,3]^2, {k, 0, n-1}]/n^2, {n, 1, 30}] (* G. C. Greubel, Jan 23 2019 *)

# prepends a(0) = 0
def A178808List(size: int) -> list[int]:
    A358387 = A358387gen()
    A358388 = A358388gen()
    return [(next(A358387) - next(A358388)) // 4 for n in range(size)]
print(A178808List(18)) # Peter Luschny, Nov 15 2022

a(n) ~ (1 + sqrt(2))^(4*n) / (16*Pi*n^2). - Vaclav Kotesovec, Jan 24 2019
G.f.: Integral(hypergeom([1/2, 1/2], [2], -32*x/(1 - 34*x + x^2))/((1 - x)*(1 - 34*x + x^2)^(1/2))). - Mark van Hoeij, Nov 10 2022
a(n) = (6*A001850(n)*A001850(n-1) - A001850(n)^2 - A001850(n-1)^2)/8. - Mark van Hoeij, Nov 12 2022
a(n) = (3*f(n)*f(n-1) - g(n))/4, where g(n) = hypergeom([n, -n, 1/2], [1, 1], -8) and f(n) = hypergeom([-n, -n], [1], 2). This formula also gives an integer value for n = 0. - Peter Luschny, Nov 13 2022

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

A358387 a(n) = 3 * h(n - 1) * h(n) for n >= 1, where h(n) = hypergeom([-n, -n], [1], 2), and a(0) = 1.

Original entry on oeis.org

1, 9, 117, 2457, 60669, 1620729, 45385461, 1311647913, 38774378493, 1165936210281, 35529105456117, 1094291069720121, 34000718751227133, 1064200845293945433, 33516300131277352821, 1061218377653812515657, 33757038339556757274621, 1078167326486278065165513

Offset: 0

Peter Luschny, Nov 15 2022

nonn

Cf. A358388.

h := n -> hypergeom([-n, -n], [1], 2):
A358387 := n -> ifelse(n = 0, 1, 3*h(n-1)*h(n)):
seq(simplify(A358387(n)), n = 0..17);

def A358387gen() -> Generator:
    b, a, n = 1, 3, 1
    yield b
    while True:
        yield  3 * a * b
        n += 1
        aa = a * (6 * n - 3)
        bb = b * (n - 1)
        b, a = a, (aa - bb) // n
A358387 = A358387gen()
print([next(A358387) for n in range(18)])

a(n) ~ 3*sqrt(2) * (1 + sqrt(2))^(4*n) / (8*Pi*n). - Vaclav Kotesovec, Jan 08 2024

cp's OEIS Frontend

A178808 a(n) = (1/n^2) * Sum_{k = 0..n-1} (2k+1)(D_k)^2, where D_0, D_1, ... are central Delannoy numbers given by A001850.

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

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A358387 a(n) = 3 * h(n - 1) * h(n) for n >= 1, where h(n) = hypergeom([-n, -n], [1], 2), and a(0) = 1.

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cp's OEIS Frontend

A178808 a(n) = (1/n^2) * Sum_{k = 0..n-1} (2*k+1)*(D_k)^2, where D_0, D_1, ... are central Delannoy numbers given by A001850.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A358387 a(n) = 3 * h(n - 1) * h(n) for n >= 1, where h(n) = hypergeom([-n, -n], [1], 2), and a(0) = 1.

Views

Author

Keywords

Crossrefs

Programs

Maple

Python

Formula

A178808 a(n) = (1/n^2) * Sum_{k = 0..n-1} (2k+1)(D_k)^2, where D_0, D_1, ... are central Delannoy numbers given by A001850.