A110170 -id:A110170 - cp's OEIS frontend

A277919 Triangle read by rows: CL(n,k) is the number of labeled subgraphs with k edges of the n-cycle C_n.

1, 1, 1, 3, 2, 1, 7, 6, 3, 1, 15, 16, 10, 4, 1, 31, 40, 30, 15, 5, 1, 63, 96, 84, 50, 21, 6, 1, 127, 224, 224, 154, 77, 28, 7, 1, 255, 512, 576, 448, 258, 112, 36, 8, 1, 511, 1152, 1440, 1248, 810, 405, 156, 45, 9, 1, 1023, 2560, 3520, 3360, 2420, 1362, 605, 210, 55, 10, 1

Offset: 0

John P. McSorley, Nov 03 2016

			For row 3 of the triangle below: there are 7 labeled subgraphs of the triangle C_3 with 0 edges, 6 with 1 edge, 3 with 2 edges, and 1 with 3 edges (C_3 itself).
Triangle begins:
     1;
     1,    1;
     3,    2,    1;
     7,    6,    3,    1;
    15,   16,   10,    4,    1;
    31,   40,   30,   15,    5,    1;
    63,   96,   84,   50,   21,    6,   1;
   127,  224,  224,  154,   77,   28,   7,   1;
   255,  512,  576,  448,  258,  112,  36,   8,  1;
   511, 1152, 1440, 1248,  810,  405, 156,  45,  9,  1;
  1023, 2560, 3520, 3360, 2420, 1362, 605, 210, 55, 10, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274
Thomas Selig, Combinatorial aspects of sandpile models on wheel and fan graphs, arXiv:2202.06487 [math.CO], 2022.

Row sums give A005592.

Middle diagonal gives A110170.

T(n)={[Vecrev(p) | p<-Vec((1 - 2*x + 2*x^2)/((1-x)*(1 - y*x - 2*x + y*x^2)) + O(x*x^n))]}
{ my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Sep 27 2019

The identity CL(n,k) = 2^(n-2*k) * CL(n,n-k) can be proved combinatorially.

G.f.: (1 - 2*x + 2*x^2)/((1-x)*(1 - y*x - 2*x + y*x^2)). - Andrew Howroyd, Sep 27 2019

More terms from John P. McSorley, Nov 17 2016

A361817 Expansion of 1/sqrt(1 - 4x(1-x)^4).

Original entry on oeis.org

1, 2, -2, -16, -10, 118, 304, -500, -3754, -2488, 30866, 83716, -135568, -1080972, -792876, 9090484, 25788118, -39325156, -335074520, -271779024, 2820643842, 8348113120, -11788972644, -107836934448, -96107852032, 900943403012, 2778574561276, -3596374190416

Offset: 0

Seiichi Manyama, Mar 25 2023

sign

Cf. A085362, A110170, A162478, A359489, A359758, A360132, A361815, A361816.

Cf. A361813.

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1-x)^4))

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k,k) * binomial(4*k,n-k).

n*a(n) = 2 * ( (2*n-1)*a(n-1) - 4*(2*n-2)*a(n-2) + 6*(2*n-3)*a(n-3) - 4*(2*n-4)*a(n-4) + (2*n-5)*a(n-5) ) for n > 4.

A382332 Expansion of 1/(1 - 4*x/(1-x)^2)^(7/2).

Original entry on oeis.org

1, 14, 154, 1470, 12866, 106078, 837018, 6385262, 47420674, 344553902, 2458367898, 17272647966, 119770278978, 821068784382, 5572735854234, 37490757508302, 250247764120578, 1658681038111566, 10924592141535898, 71541334475749502, 466060971286552642

Offset: 0

Seiichi Manyama, Mar 30 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..600

Cf. A110170, A377198, A382274.

Cf. A020918, A377200.

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/(1 - 4*x/(1-x)^2)^(7/2))); // Vincenzo Librandi, May 12 2025

Table[Sum[(-4)^k* Binomial[-7/2,k]*Binomial[n+k-1, n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, May 12 2025 *)

a(n) = sum(k=0, n, (-4)^k*binomial(-7/2, k)*binomial(n+k-1, n-k));

a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (7-5*k/n) * (n-k) * a(k).
a(n) = ((7*n+7)*a(n-1) - (7*n-28)*a(n-2) + (n-3)*a(n-3))/n for n > 2.
a(n) = Sum_{k=0..n} (-4)^k * binomial(-7/2,k) * binomial(n+k-1,n-k).
a(n) = 14*n*hypergeom([9/2, 1-n, 1+n], [3/2, 2], -1) for n > 0. - Stefano Spezia, Mar 30 2025
a(n) ~ 2^(5/4) * (1 + sqrt(2))^(2*n) * n^(5/2) / (15*sqrt(Pi)). - Vaclav Kotesovec, May 03 2025

A110169 Triangle read by rows: T(n,k) (0<=k<=n) is the number of Delannoy paths of length n that start with exactly k (1,1) steps.

Original entry on oeis.org

1, 2, 1, 10, 2, 1, 50, 10, 2, 1, 258, 50, 10, 2, 1, 1362, 258, 50, 10, 2, 1, 7306, 1362, 258, 50, 10, 2, 1, 39650, 7306, 1362, 258, 50, 10, 2, 1, 217090, 39650, 7306, 1362, 258, 50, 10, 2, 1, 1196834, 217090, 39650, 7306, 1362, 258, 50, 10, 2, 1, 6634890, 1196834

Offset: 0

Emeric Deutsch, Jul 14 2005

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).

Row sums are the central Delannoy numbers (A001850). Column 0 yields A110170 (first differences of the central Delannoy numbers). sum(k*T(n,k),k=0..n)=A089165(n-1) (n>=1; partial sums of the central Delannoy numbers).

			T(3,2)=2 because we have DDNE and DDEN.
Triangle starts:
1;
2,1;
10,2,1;
50,10,2,1;
258,50,10,2,1;

Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

Cf. A001850, A110170, A089165.

Maple
```
with(orthopoly): S:=proc(n,k) if k
				
```

T(n, k) = A001850(n-k)-A001850(n-k-1) for k

T(n, k) = P_{n-k}(3)-P_{n-k-1}(3) for k

G.f.: (1-z)/((1-t*z)*sqrt(1-6*z+z^2)).

Keyword tabf changed to tabl by Michel Marcus, Apr 09 2013

A376810 Expansion of 1/sqrt(1 - 4*x/(1 - x^2)^2).

Original entry on oeis.org

1, 2, 6, 24, 94, 378, 1544, 6380, 26598, 111658, 471358, 1998924, 8509368, 36341278, 155634228, 668116136, 2874157222, 12387209982, 53475080494, 231189987224, 1000834283190, 4337864724462, 18821884379924, 81748960355484, 355383570351664, 1546239230878154

Offset: 0

Seiichi Manyama, Oct 04 2024

nonn

Cf. A110170, A376811.

Cf. A349713.

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1-x^2)^2))

a(n) = sum(k=0, n\2, binomial(2*n-3*k-1, k)*binomial(2*n-4*k, n-2*k));

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k-1,k) * binomial(2*n-4*k,n-2*k).

A376811 Expansion of 1/sqrt(1 - 4*x/(1 - x^3)^2).

Original entry on oeis.org

1, 2, 6, 20, 74, 276, 1044, 3998, 15450, 60128, 235332, 925332, 3652508, 14464490, 57442074, 228670140, 912239782, 3646027752, 14596600800, 58523194734, 234954663396, 944418233612, 3800327339532, 15307785490560, 61716607166724, 249033637247898, 1005661821858414

Offset: 0

Seiichi Manyama, Oct 04 2024

nonn

Cf. A110170, A376810.

Cf. A376791.

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1-x^3)^2))

a(n) = sum(k=0, n\3, binomial(2*n-5*k-1, k)*binomial(2*n-6*k, n-3*k));

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-5*k-1,k) * binomial(2*n-6*k,n-3*k).

A382274 Expansion of 1/(1 - 4*x/(1-x)^2)^(5/2).

Original entry on oeis.org

1, 10, 90, 730, 5570, 40762, 289370, 2007210, 13671170, 91750250, 608294490, 3991833210, 25968131010, 167664187290, 1075453670490, 6858654320970, 43517809896450, 274862176368330, 1728960219827290, 10835520927931930, 67679638209628098, 421442759107879930

Offset: 0

Seiichi Manyama, Mar 29 2025

nonn

Cf. A110170, A377198, A382332.

Cf. A002802, A377199.

a(n) = sum(k=0, n, (-4)^k*binomial(-5/2, k)*binomial(n+k-1, n-k));

a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (5-3*k/n) * (n-k) * a(k).
a(n) = ((7*n+3)*a(n-1) - (7*n-24)*a(n-2) + (n-3)*a(n-3))/n for n > 2.
a(n) = Sum_{k=0..n} (-4)^k * binomial(-5/2,k) * binomial(n+k-1,n-k).
a(n) = 10*n*hypergeom([7/2, 1-n, 1+n], [3/2, 2], -1) for n > 0. - Stefano Spezia, Mar 30 2025
a(n) ~ 2^(3/4) * n^(3/2) * (1 + sqrt(2))^(2*n) / (3*sqrt(Pi)). - Vaclav Kotesovec, Apr 13 2025

A376809 Expansion of 1/sqrt(1 - 4*x^3/(1 - x)^2).

Original entry on oeis.org

1, 0, 0, 2, 4, 6, 14, 34, 72, 154, 346, 774, 1714, 3822, 8574, 19238, 43204, 97254, 219286, 494962, 1118502, 2530522, 5730762, 12989634, 29467718, 66901378, 151996338, 345556218, 786092266, 1789284762, 4074927962, 9284968682, 21166439112, 48273612954, 110142596298

Offset: 0

Seiichi Manyama, Oct 04 2024

Partial sums are A098479.

Cf. A025178, A110170.

my(N=40, x='x+O('x^N)); Vec(1/sqrt(1-4*x^3/(1-x)^2))

a(n) = sum(k=0, n\3, binomial(2*k,k) * binomial(n-k-1,n-3*k));

a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(n-k-1,n-3*k).

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A277919 Triangle read by rows: CL(n,k) is the number of labeled subgraphs with k edges of the n-cycle C_n.

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A361817 Expansion of 1/sqrt(1 - 4*x*(1-x)^4).

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A382332 Expansion of 1/(1 - 4*x/(1-x)^2)^(7/2).

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A110169 Triangle read by rows: T(n,k) (0<=k<=n) is the number of Delannoy paths of length n that start with exactly k (1,1) steps.

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A376810 Expansion of 1/sqrt(1 - 4*x/(1 - x^2)^2).

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A376811 Expansion of 1/sqrt(1 - 4*x/(1 - x^3)^2).

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A382274 Expansion of 1/(1 - 4*x/(1-x)^2)^(5/2).

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A376809 Expansion of 1/sqrt(1 - 4*x^3/(1 - x)^2).

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A361817 Expansion of 1/sqrt(1 - 4x(1-x)^4).