A108958 -id:A108958 - cp's OEIS frontend

A005317 a(n) = (2^n + C(2*n,n))/2.

1, 2, 5, 14, 43, 142, 494, 1780, 6563, 24566, 92890, 353740, 1354126, 5204396, 20066492, 77575144, 300572963, 1166868646, 4537698722, 17672894044, 68923788698, 269129985796, 1052051579012, 4116719558104, 16123810230158, 63205319996092, 247959300028484

Offset: 0

N. J. A. Sloane and Peter Fishburn

Hankel transform is A008619. - Paul Barry, Nov 13 2007

a(n) is the number of lattice paths from (0,0) to (n,n) using E(1,0) and N(0,1) as steps that horizontally cross the diagonal y = x with even many times. For example, a(2) = 5 because there are 6 paths in total and only one of them horizontally crosses the diagonal with odd many times, namely, NEEN. - Ran Pan, Feb 01 2016

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1664 (first 179 terms from Vincenzo Librandi)
Jelena Đokic, A short note on the order of the double reduced 2-factor transfer digraph for rectangular grid graphs, arXiv:2308.04155 [math.CO], 2023.
Jelena Đokić, Olga Bodroža-Pantić, and Ksenija Doroslovački, A spanning union of cycles in rectangular grid graphs, thick grid cylinders and Moebius strips, Transactions on Combinatorics (2023) Art. 27132.
T. Kløve, Generating functions for the number of permutations with limited displacement, Electron. J. Combin., 16 (2009), #R104. - From _N. J. A. Sloane_, May 04 2011.
Peter Fishburn, Letter to N. J. A. Sloane, Mar 1987
Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
Simon Plouffe, Approximations of generating functions and a few conjectures, arXiv:0911.4975 [math.NT], 2009.

Cf. A008619, A108958.

[(2^n+Binomial(2*n,n))/2: n in [0..26]];  // Bruno Berselli, Jun 20 2011

Maple
```
f := n->(2^n+binomial(2*n,n))/2;
```

Table[(2^n + Binomial[2 n, n])/2, {n, 0, 26}] (* Michael De Vlieger, Feb 01 2016 *)

makelist(sum((-1)^k*binomial(2*n,n-2*k),k,0,floor(n/2)),n,0,26); /* Bruno Berselli, Jun 20 2011 */

a(n)=(2^n+binomial(2*n,n))/2 \\ Charles R Greathouse IV, Dec 20 2011

From Simon Plouffe, Feb 18 2011: (Start)
G.f.: (1/2)*(-4*x+1+(-(4*x-1)*(2*x-1)^2)^(1/2))/(4*x-1)/(2*x-1).
Recurrence: 0 = (-24-28*n-8*n^2)*a(n+1) + (18+22*n+6*n^2)*a(n+2) + (-3-4*n-n^2)*a(n+3), a(0)=1, a(1)=2, a(2)=5. (End)
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*C(2*n, n-2*k), n > 0. - Mircea Merca, Jun 20 2011
E.g.f.: exp(2*x)*(1+BesselI(0,2*x))/2 = G(0)/2; G(k) = 1 + (k)!/(P-2*x*(2*k+1)*(P^2)/(2*x*(2*k+1)*P+(k+1)^2*k!/G(k+1))), where P:=((2*k)!)/(2^k)/((k)!) (continued fraction). - Sergei N. Gladkovskii, Dec 20 2011
a(n) = Sum_{r=0..n} k*(k+1)/2 where k=C(n,r). - J. M. Bergot, Sep 04 2013
a(n) = binomial(2*n,n) - A108958(n). - Ran Pan, Feb 01 2016
a(n) ~ 2^(2*n-1)/sqrt(n*Pi). - Stefano Spezia, Apr 17 2024

A143418 Triangle read by rows. T(n,k) = binomial(n,k)*(binomial(n,k)-1)/2.

Original entry on oeis.org

1, 3, 3, 6, 15, 6, 10, 45, 45, 10, 15, 105, 190, 105, 15, 21, 210, 595, 595, 210, 21, 28, 378, 1540, 2415, 1540, 378, 28, 36, 630, 3486, 7875, 7875, 3486, 630, 36, 45, 990, 7140, 21945, 31626, 21945, 7140, 990, 45, 55, 1485, 13530, 54285, 106491, 106491

Offset: 1

Gary W. Adamson and Roger L. Bagula, Aug 14 2008

Row sums = A108958: (1, 6, 27, 110, 430, 1652, ...).

			Row 4 of Pascal's triangle (1, 4, 6, 4, 1) with each term squared = (1, 16, 36, 16, 1), then subtracting (1, 4, 6, 4, 1) = (0, 12, 30, 12, 0). Dividing by 2 and deleting the zeros, we get row 4 of A143418: (6, 15, 6).
First few rows of the triangle =
1;
3, 3;
6, 15, 6;
10, 45, 45, 10;
15, 105, 190, 105, 15;
21, 210, 595, 595, 210, 21;
28, 378, 1540, 2415, 1540, 378, 28;
...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007318, A108958.

A143418 := proc(n,k)
        binomial(n,k)*(binomial(n,k)-1)/2 ;
end proc:
seq(seq(A143418(n,k),k=1..n-1),n=1..12) ; # R. J. Mathar, Apr 04 2012

Table[Binomial[n,k] (Binomial[n,k]-1)/2,{n,20},{k,n-1}]//Flatten (* Harvey P. Dale, Jun 14 2021 *)

T(n,k) = A065420(n-1,k-1)/2. - R. J. Mathar, Apr 04 2012

Corrected by Harvey P. Dale, Jun 14 2021

A143420 Row sums of triangle A373101.

Original entry on oeis.org

1, 8, 55, 370, 2520, 17472, 123151, 880070, 6360706, 46402312, 341153384, 2524722928, 18789734496, 140521154048, 1055383259791, 7956220907758, 60179579570382, 456545145078408, 3472804505717170

Offset: 2

Gary W. Adamson, Aug 14 2008

Each term in the sequence is a sum of tetrahedral numbers.

The underlying triangle mentioned as A143419 was lost and is now restored in A373101. - Georg Fischer, May 23 2024

			a(5) = 370 = (20 + 165 + 165 + 20) = C(6,3) + C(11,3) + C(11,8) + C(6,3).

Cf. A108958 (row sums of A143418), A373101.

seq(add((binomial(n,k)^3 - binomial(n,k))/6,k=1..n-1),n=2..20); # Georg Fischer, May 23 2024

Definition changed, a(7) corrected and more terms from Georg Fischer, May 23 2024

cp's OEIS Frontend

A005317 a(n) = (2^n + C(2*n,n))/2.

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A143418 Triangle read by rows. T(n,k) = binomial(n,k)*(binomial(n,k)-1)/2.

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A143420 Row sums of triangle A373101.

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