A001850 -id:A001850 - cp's OEIS frontend

A110170 First differences of the central Delannoy numbers (A001850).

1, 2, 10, 50, 258, 1362, 7306, 39650, 217090, 1196834, 6634890, 36949266, 206549250, 1158337650, 6513914634, 36718533570, 207412854786, 1173779487810, 6653482333450, 37770112857074, 214694383882498, 1221832400430482, 6961037946938250, 39697830840765090, 226596964146630658

Offset: 0

Emeric Deutsch, Jul 14 2005

Number of Delannoy paths of length n that do not start with a (1, 1) step (a Delannoy path of length n is a path from (0, 0) to (n, n), consisting of steps E = (1, 0), N = (0, 1) and D = (1, 1)). Example: a(1) = 2 because we have NE and EN. Column 0 of A110169 (also nonzero entries in each column of A110169).

For n > 0: a(n) = A128966(2*n,n). - Reinhard Zumkeller, Jul 20 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Thomas Selig, Combinatorial aspects of sandpile models on wheel and fan graphs, arXiv:2202.06487 [math.CO], 2022.
Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

Cf. A001850, A110169.

Cf. A085362, A162478, A359758, A360132.

a110170 0 = 1
a110170 n = a128966 (2 * n) n  -- Reinhard Zumkeller, Jul 20 2013

with(orthopoly): a:=proc(n) if n=0 then 1 else P(n,3)-P(n-1,3) fi end: seq(a(n),n=0..25);
a := n -> `if`(n=0, 1, 2*hypergeom([1 - n, -n], [1], 2)):
seq(simplify(a(n)), n=0..24); # Peter Luschny, May 22 2017

CoefficientList[Series[(1 - x)/Sqrt[1 - 6 * x + x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)

x='x+O('x^66); Vec((1-x)/sqrt(1-6*x+x^2)) \\ Joerg Arndt, May 16 2013

G.f.: (1-z)/sqrt(1-6*z+z^2).
a(n) = P_n(3) - P_{n-1}(3) (n >= 1), where P_j is j-th Legendre polynomial.
From Paul Barry, Oct 18 2009: (Start)
G.f.: (1-x)/(1-x-2x/(1-x-x/(1-x-x/(1-x-x/(1-... (continued fraction);
G.f.: 1/(1-2x/((1-x)^2-x/(1-x/((1-x)^2-x/(1-x/((1-x)^2-x/(1-... (continued fraction);
a(n) = Sum_{k = 0..n} (0^(n + k) + C(n + k - 1, 2k - 1)) * C(2k, k) = 0^n + Sum_{k = 0..n} C(n + k - 1, 2k - 1) * C(2k, k). (End)
D-finite with recurrence: n*(2*n-3)*a(n) = 2*(6*n^2-12*n+5)*a(n-1) - (n-2)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 2^(-1/4)*(3+2*sqrt(2))^n/sqrt(Pi*n). - Vaclav Kotesovec, Oct 18 2012
a(n) = A277919(2n,n). - John P. McSorley, Nov 23 2016
a(n) = 2*hypergeom([1 - n, -n], [1], 2) for n>0. - Peter Luschny, May 22 2017
D-finite with recurrence: n*a(n) +(-7*n+5)*a(n-1) +(7*n-16)*a(n-2) +(-n+3)*a(n-3)=0. - R. J. Mathar, Jan 15 2020
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (n^2-k^2) * a(k). - Seiichi Manyama, Mar 28 2023
G.f.: Sum_{n >= 0} binomial(2*n, n)*x^n/(1 - x)^(2*n) = 1 + 2*x + 10*x^2 + 50*x^3 + .... - Peter Bala, Apr 17 2024

A243949 Squares of the central Delannoy numbers: a(n) = A001850(n)^2.

Original entry on oeis.org

1, 9, 169, 3969, 103041, 2832489, 80802121, 2365752321, 70611901441, 2139090528969, 65568745087209, 2029206892664961, 63300531617048961, 1987912809986437161, 62787371136571152009, 1992942254830520803329, 63531842302018973818881, 2033004661359005674887561

Offset: 0

Paul D. Hanna, Aug 17 2014

In general, we have the binomial identity:
if b(n) = Sum_{k=0..n} t^k * C(2*k, k) * C(n+k, n-k), then b(n)^2 = Sum_{k=0..n} (t^2+t)^k * C(2*k, k)^2 * C(n+k, n-k), where the g.f. of b(n) is 1/sqrt(1 - (4*t+2)*x + x^2), and the g.f. of b(n)^2 is 1 / AGM(1-x, sqrt((1+x)^2 - (4*t+2)^2*x)), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.
Note that the g.f. of A001850 is 1/sqrt(1 - 6*x + x^2).
Limit_{n -> oo} a(n+1)/a(n) = (3 + 2*sqrt(2))^2 = 17 + 12*sqrt(2).
From Gheorghe Coserea, Jul 05 2016: (Start)
Diagonal of the rational function 1/(1 - x - y - z - x*y + x*z - y*z - x*y*z).
Annihilating differential operator: x*(x-1)*(x+1)*(x^2-34*x+1)*Dx^2 + (3*x^4-66*x^3-70*x^2+70*x-1)*Dx + x^3-7*x^2-35*x+9.
(End).
The sequence b(n) mentioned above is the sequence of shifted Legendre polynomials P(n,2*t + 1) (see A063007). See Zudilin for a g.f. for the sequence b(n)^2. - Peter Bala, Mar 02 2017

			G.f.: A(x) = 1 + 9*x + 169*x^2 + 3969*x^3 + 103041*x^4 + 2832489*x^5 +...

Seiichi Manyama, Table of n, a(n) for n = 0..500
A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"
W. Zudilin, A generating function of the squares of Legendre polynomials, arXiv:1210.2493v2 [math.CA], 2012.

Sequences of the form LegendreP(n, 2*m+1)^2: A000012 (m=0), this sequence (m=1), A243943 (m=2), A243944 (m=3), A243007 (m=4).
Related to diagonal of rational functions: A268545 - A268555.
Cf. A001850, A243945, A245925.

[Evaluate(LegendrePolynomial(n), 3)^2 : n in [0..40]]; // G. C. Greubel, May 17 2023

Table[Sum[2^k *Binomial[2*k, k]^2 *Binomial[n+k, n-k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Aug 18 2014 *)
a[n_]:= HypergeometricPFQ[{1/2, -n, n+1}, {1, 1}, -8];
Table[a[n], {n, 0, 17}] (* Peter Luschny, Mar 14 2018 *)
LegendreP[Range[0, 30], 3]^2 (* G. C. Greubel, May 17 2023 *)

{a(n) = sum(k=0, n, 2^k * binomial(2*k, k)^2 * binomial(n+k, n-k) )}
for(n=0, 20, print1(a(n), ", "))

{a(n)=polcoeff( 1 / agm(1-x, sqrt((1+x)^2 - 36*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

from math import comb
def A243949(n): return sum(comb(n,k)*comb(n+k,k) for k in range(n+1))**2 # Chai Wah Wu, Mar 23 2023

[gen_legendre_P(n,0,3)^2 for n in range(41)] # G. C. Greubel, May 17 2023

G.f.: 1 / AGM(1-x, sqrt(1-34*x+x^2)). - Paul D. Hanna, Aug 30 2014
a(n) = Sum_{k=0..n} 2^k * C(2*k, k)^2 * C(n+k, n-k).
a(n)^(1/2) = Sum_{k=0..n} C(2*k, k) * C(n+k, n-k).
Recurrence: n^2*(2*n-3)*a(n) = (2*n-1)*(35*n^2 - 70*n + 26)*a(n-1) - (2*n-3)*(35*n^2 - 70*n + 26)*a(n-2) + (n-2)^2*(2*n-1)*a(n-3). - Vaclav Kotesovec, Aug 18 2014
a(n) ~ (4 + 3*sqrt(2)) * (3 + 2*sqrt(2))^(2*n) / (8*Pi*n). - Vaclav Kotesovec, Aug 18 2014
From Gheorghe Coserea, Jul 05 2016: (Start)
G.f.: hypergeom([1/12, 5/12],[1],27648*x^4*(x^2-34*x+1)*(x-1)^2/(1-36*x+134*x^2-36*x^3+x^4)^3)/(1-36*x+134*x^2-36*x^3+x^4)^(1/4).
0 = x*(x-1)*(x+1)*(x^2-34*x+1)*y'' + (3*x^4-66*x^3-70*x^2+70*x-1)*y' + (x^3-7*x^2-35*x+9)*y, where y is g.f.
(End)
a(n) = Sum_{k = 0..n} 4^k*binomial(n+k,2*k)^2*binomial(2*k,k). - Peter Bala, Mar 02 2017
a(n) = hypergeom([1/2, -n, n + 1], [1, 1], -8). - Peter Luschny, Mar 14 2018
G.f.: Sum_{n >= 0} (2^n)*binomial(2*n,n)^2 *x^n/(1-x)^(2*n+1). - Peter Bala, Feb 07 2022

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.

Original entry on oeis.org

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

A248136 Least positive integer m such that m + n divides D(m) + D(n), where D(.) is given by A001850.

Original entry on oeis.org

1, 20, 3, 6, 1, 4, 200, 299, 5, 29, 4, 119, 5, 61, 3, 3, 6, 64, 31, 2, 21, 35, 6, 2974, 17, 1052, 27, 109, 10, 4, 3, 50, 65, 177, 22, 29, 5, 25, 15, 29, 29, 584, 83, 163, 9, 152, 19, 19, 29, 32, 15, 35, 4, 25, 239, 1122, 185, 76, 35, 97

Offset: 1

Zhi-Wei Sun, Oct 02 2014

nonn

Conjecture: a(n) exists for any n > 0.

			a(2) = 20 since 2 + 20 = 22 divides D(2) + D(20) = 13 + 260543813797441 = 260543813797454 = 22*11842900627157.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A001850, A247824, A247937, A247940, A248124, A248125, A248133, A248137, A248139, A248142.

d[n_]:=Sum[Binomial[n,k]Binomial[n+k,k],{k,0,n}]
Do[m=1;Label[aa];If[Mod[d[m]+d[n],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]

A089165 Partial sums of the central Delannoy numbers (A001850).

Original entry on oeis.org

1, 4, 17, 80, 401, 2084, 11073, 59712, 325441, 1788004, 9885457, 54932176, 306528145, 1716461764, 9640310017, 54282691840, 306337928449, 1732172652868, 9811489710737, 55660919625680, 316204733423121, 1798580947651044

Offset: 0

Emeric Deutsch, Dec 06 2003

nonn

Number of peaks at odd level in all Schroeder paths (i.e., consisting of steps U=(1,1),D=(1,-1),H=(2,0) and never going below the x-axis) from (0,0) to (2n+2,0). Example: a(1)=4 because HH,HU*D,U*DH,UHD,U*DU*D,UUDD contain 4 peaks at odd level (indicated by *).
From Evert-Jan D. Pol (evert-jan.pol(AT)philips.com), Oct 25 2005: "Also appears in the context of infinite lattices of unit resistors. The paper by Atkinson and van Steenwijk shows how to calculate the resulting resistance R(n,p) between two nodes in the lattice that are apart by the vector (n,p). The resulting values can be written in the form r+s/Pi, where r and s are rational numbers.
"Here we concentrate on the rational part r. The paper gives values for a single quadrant in the integer plane. Other quadrants can be filled with mirror-images of the given quadrant. Casual inspection of the values shows that the numbers are most easily analysed by looking at the diagonals (n+k,n-k) for n=0,1,2,... and fixed k. The rational part of the values on these diagonals appears to be a polynomial sequence of degree 2k-1, apart from the alternating sign.
"Similarly, the absolute value of the rational part of the values on the diagonals (n+k+1,n-k) is a polynomial sequence of degree 2k. Assuming these observations to be true, the entire plane of rational values can be constructed from the single sequence R(0,p)! The values off the axes are simply extrapolated from values on and closer to the axes based on the polynomial form of the diagonals, with the proper sign. The sequence R(0,p) begins with 0, 1/2, 2-4/Pi, 17/2-24/Pi, 40-368/(3Pi) and twice the rational part of this sequence is A089165. The mathematica program given here is copied verbatim from the paper."
The first Mathematica code produces 0, 1, 4 - 8/Pi, 17 - 48/Pi, 80 - 736/(3*Pi), 401 - 3760/(3*Pi), 2084 - 98104/(15*Pi), 11073 - 521696/(15*Pi), 59712 - 19696256/(105*Pi), 325441 - 7156768/(7*Pi), 1788004 - 1769409304/(135*Pi); ... and the integer part gives the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. Atkinson and F. J. van Steenwijk, Infinite Resistive Lattices.
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.

Cf. A001850, A006318.

G:=1/((1-z)*sqrt(1-6*z+z^2)): Gser:=series(G,z=0,26): seq(coeff(Gser,z,n),n=0..23); # Emeric Deutsch, May 05 2006

alphas[beta_]:=Log[2-Cos[beta]+Sqrt[3+Cos[beta]*(Cos[beta]-4)]]; Rsqu[n_, p_]:=Simplify[(1/Pi)*Integrate[(1-Exp[ -Abs[n]*alphas[beta]]*Cos[p*beta])/Sinh[alphas[beta]], {beta, 0, Pi}]]; Table[Expand[2Rsqu[0, k]], {k, 0, 8}] (Evert-Jan D. Pol)
f[n_] := Sum[ Binomial[i, j] Binomial[j, i-j], {i, 0, 2n}, {j, 0, n}]; Table[ f@n, {n, 0, 21}] (* or *)
CoefficientList[ Series[ 1/((1 - x)Sqrt[1 - 6x + x^2]), {x, 0, 21}], x] (* Robert G. Wilson v, May 04 2006 *)

a(n)=sum(i=0,2*n,sum(j=0,n,binomial(i,j)*binomial(j,i-j)))

x + O(x^66); Vec(deriv(atan(x*(1-x-(1-6*x+x^2)^(1/2))/(2*x)))) \\ Joerg Arndt, Apr 21 2011

G.f.: 1/((1-z)*sqrt(1-6*z+z^2)).
a(n) = Sum_{j=0..n} Sum_{i=0..2n} binomial(i, j)*binomial(j, i-j). - Benoit Cloitre, Oct 23 2004
a(n) = Sum_{k=0..n} C(n+k+1,2k+1)*A000984(k). - Paul Barry, Jun 03 2009
G.f.: d/dx atan(x*A006318(x)). - Vladimir Kruchinin
Recurrence: n*a(n) = (7*n-3)*a(n-1) - (7*n-4)*a(n-2) + (n-1)*a(n-3). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ sqrt(48+34*sqrt(2))*(3+2*sqrt(2))^n/(8*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012

A137219 a(n) = (A126086(n) - 3*A001850(n) + 2)/6.

Original entry on oeis.org

1, 62, 2649, 116360, 5364701, 256452714, 12582472897, 629389744448, 31955247002601, 1641724953315062, 85159811841234857, 4452782349569991736, 234393562418967430389, 12409423916979629786322, 660253088667210584565249

Offset: 1

Vladeta Jovovic, Mar 06 2008, Mar 16 2008

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A001850, A047665, A126086, A137220.

A126086 := proc(n) local x,y,z ; coeftayl(coeftayl(coeftayl(1/(1-x-y-z-x*y-x*z-y*z-x*y*z),z=0,n),y=0,n),x=0,n) ; end: A001850 := proc(n) local k ; add(binomial(n,k)*binomial(n+k,k),k=0..n) ; end: A137219 := proc(n) (A126086(n)-3*A001850(n)+2)/6 ; end: seq(A137219(n),n=1..30) ; # R. J. Mathar, Apr 01 2008

A126086[n_]:= A126086[n]= Sum[(-1)^k*Binomial[n+k,n]*HypergeometricPFQ[{-k, n+1, n+1}, {1,1}, 1], {k,0,2*n}];
A001850[n_]:= Hypergeometric2F1[-n, n+1, 1, -1];
A137219[n_]:= (A126086[n] - 3*A001850[n] + 2)/6;
Table[A137219[n], {n, 30}] (* G. C. Greubel, Jan 05 2022 *)

def A137219(n): return round( sum( binomial(binomial(j, n), 3)/2^(j+1) for j in (0..1000) ) )
[A137219(n) for n in (1..30)] # G. C. Greubel, Jan 05 2022

a(n) = Sum_{m >= 0} binomial(binomial(m, n), 3)/2^(m+1).

More terms from R. J. Mathar, Apr 01 2008

A137220 a(n) = (A126086(n) + 3*A001850(n) + 2)/6.

Original entry on oeis.org

1, 4, 75, 2712, 116681, 5366384, 256461703, 12582521536, 629390010177, 31955248465164, 1641724961412515, 85159811886281576, 4452782349821587705, 234393562420377364008, 12409423916987553634575, 660253088667255226947072

Offset: 0

Vladeta Jovovic, Mar 06 2008, Mar 16 2008

Andrew Howroyd, Table of n, a(n) for n = 0..200

Column k=3 of A330942.

Cf. A047665, A137219.

A126086 := proc(n) local x,y,z ; coeftayl(coeftayl(coeftayl(1/(1-x-y-z-x*y-x*z-y*z-x*y*z),z=0,n),y=0,n),x=0,n) ; end: A001850 := proc(n) local k ; add(binomial(n,k)*binomial(n+k,k),k=0..n) ; end: A137220 := proc(n) (A126086(n)+3*A001850(n)+2)/6 ; end: seq(A137220(n),n=0..30) ; # R. J. Mathar, Apr 01 2008

T[n_, k_] := With[{m = n k}, Sum[Binomial[Binomial[j, n] + k - 1, k] Sum[ (-1)^(i - j) Binomial[i, j], {i, j, m}], {j, 0, m}]];
Table[T[n, 3], {n, 0, 15}] (* Jean-François Alcover, Apr 10 2020, after Andrew Howroyd *)

a(n) = {sum(j=0, 3*n, binomial(binomial(j,n)+2, 3) * sum(i=j, 3*n, (-1)^(i-j)*binomial(i,j)))} \\ Andrew Howroyd, Feb 09 2020

@CachedFunction
def A137220(n): return round( -sum( binomial(-binomial(j, n), 3)/2^(j+1) for j in (0..500) ) )
[A137220(n) for n in (0..30)] # G. C. Greubel, Jan 05 2022

a(n) = -Sum_{m>=0} binomial(-binomial(m,n),3)/2^(m+1).
a(n) = A137219(n) + A001850(n). - R. J. Mathar, Apr 01 2008
a(n) = Sum_{j=0..3*n} binomial(binomial(j,n)+2, 3) * (Sum_{i=j..3*n} (-1)^(i-j)*binomial(i,j)). - Andrew Howroyd, Feb 09 2020

More terms from R. J. Mathar, Apr 01 2008

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}]

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

A080609 Binomial transform of central Delannoy numbers A001850.

Original entry on oeis.org

1, 4, 20, 112, 664, 4064, 25376, 160640, 1027168, 6618496, 42904960, 279503360, 1828222720, 11999226880, 78984381440, 521218322432, 3447059138048, 22840932997120, 151607254267904, 1007830488424448, 6708862677274624

Offset: 0

Emanuele Munarini, Feb 26 2003

The Hankel transform (see A001906 for definition) of this sequence is A036442: 1, 4, 32, 512, 16384, ... . - Philippe Deléham, Jul 03 2005
Coefficient of x^n in (1 + 4*x + 2*x^2)^n - N-E. Fahssi, Jan 17 2008
Number of paths from (0,0) to (n,0) using only steps U=(1,1), H=(1,0) and D=(1,-1), U can have 2 colors and H can have 4 colors. - N-E. Fahssi, Jan 27 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Table[SeriesCoefficient[Series[1/Sqrt[1-8x+8x^2], {x, 0, n}], n], {n, 0, 12}]
Table[LegendreP[n, Sqrt[2]] 8^(n/2), {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)

x='x+O('x^66); Vec(1/sqrt(1-8*x+8*x^2)) \\ Joerg Arndt, May 07 2013

G.f.: 1 / sqrt( 1 - 8*x + 8*x^2 ).
a(n) = Sum_{k=0..n} binomial(n,k) * A001850(k).
E.g.f.: exp(4*x)*BesselI(0, 2*sqrt(2)*x). - Vladeta Jovovic, Mar 21 2004
Recurrence: n*a(n) = 4*(2*n-1)*a(n-1) - 8*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 13 2012
a(n) ~ sqrt(1+sqrt(2))*(4+2*sqrt(2))^n/sqrt(2*Pi*n). - Vaclav Kotesovec, Oct 13 2012
G.f.: G(0), where G(k)= 1 + 4*x*(1-x)*(4*k+1)/(2*k+1 - 2*x*(1-x)*(2*k+1)*(4*k+3)/(2*x*(1-x)*(4*k+3) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 22 2013
a(n) = LegendreP_n(sqrt(2))*8^(n/2). - Vladimir Reshetnikov, Nov 01 2015

cp's OEIS Frontend

A110170 First differences of the central Delannoy numbers (A001850).

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A243949 Squares of the central Delannoy numbers: a(n) = A001850(n)^2.

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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.

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A248136 Least positive integer m such that m + n divides D(m) + D(n), where D(.) is given by A001850.

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A089165 Partial sums of the central Delannoy numbers (A001850).

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A137219 a(n) = (A126086(n) - 3*A001850(n) + 2)/6.

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A137220 a(n) = (A126086(n) + 3*A001850(n) + 2)/6.

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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.

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