Christian Barrientos - cp's OEIS frontend

A374722 Number of nonisomorphic spanning trees of the nC_5-snake with constant distance between cutpoints.

1, 6, 24, 120, 570, 2850, 14100, 70500, 351750, 1758750, 8790000, 43950000, 219731250, 1098656250, 5493187500, 27465937500, 137329218750, 686646093750, 3433228125000, 17166140625000, 85830691406250, 429153457031250, 2145767226562500, 10728836132812500, 53644180371093750

Offset: 1

Christian Barrientos, Jul 17 2024

a(n) is the number of spanning trees of the cyclic snake formed with n copies of the cycle on 5 vertices. A cyclic snake is a connected graph whose block-cutpoint is a path and all its n blocks are isomorphic to the cycle C_m.

			For n=2, a(2)=6 because there are 6 spanning trees of 2C_5-snake
__ __ __ __ __ __ __ __, __ __ __ __|__ __ __, __ __ __ \/__ __ __,
            __              __           __
__ __ __ __|__ __, __ __ __|__ __, __ __|__ __
                           |            |__

Christian Barrientos, Graceful labelings of cyclic snakes, Ars Combin., 60 (2001), 85-96.

Index entries for linear recurrences with constant coefficients, signature (5,5,-25).

Cf. A374721.

Drop[CoefficientList[Series[x*(1 + x - 11*x^2 - 5*x^3)/((1 - 5*x)*(1 - 5*x^2)),{x,0,30}],x],1] (* Georg Fischer, Aug 09 2024 *)

for(n=1, 30, print1(if(n==1,1,(3/2)*(3* 5^(n - 2) + 5^floor((n - 2)/2))),",")) \\ Georg Fischer, Aug 09 2024

a(n) = (3/2)*(3* 5^(n - 2) + 5^floor((n - 2)/2)) for n > 1.
From Stefano Spezia, Jul 23 2024: (Start)
G.f.: x*(1 + x - 11*x^2 - 5*x^3)/((1 - 5*x)*(1 - 5*x^2)).
E.g.f.: (24 + 10*x - 9*cosh(5*x) - 15*cosh(sqrt(5)*x) - 9*sinh(5*x) - 3*sqrt(5)*sinh(sqrt(5)*x))/50. (End)

a(25) corrected by Georg Fischer, Aug 09 2024

A374721 Number of nonisomorphic spanning trees of the triangular snake nC_3.

Original entry on oeis.org

1, 3, 7, 21, 57, 171, 495, 1485, 4401, 13203, 39447, 118341, 354537, 1063611, 3189375, 9568125, 28700001, 86100003, 258286887, 774860661, 2324542617, 6973627851, 20920765455, 62762296365, 188286534801, 564859604403, 1694577750327, 5083733250981, 15251196564297, 45753589692891

Offset: 1

Christian Barrientos, Jul 17 2024

a(n) is the number of spanning trees of the cyclic snake formed with n copies of the cycle on 3 vertices. A cyclic snake is a connected graph whose block-cutpoint is a path and all its n blocks are isomorphic to the cycle C_m.

			For n=2 the a(2)=3 nonisomorphic spanning trees of 2C_3-snake are:
__ __ __ __, __\__ __, __\/__

Christian Barrientos, Graceful labelings of cyclic snakes, Ars Combin., 60 (2001), 85-96.

Paolo Xausa, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Triangular Snake Graph.
Index entries for linear recurrences with constant coefficients, signature (3,3,-9).

Cf. A374722.

A374721[n_] := 2*3^(n - 2) + 3^Floor[(n - 2)/2]; Array[A374721, 30] (* or *)
LinearRecurrence[{3, 3, -9}, {1, 3, 7}, 30] (* Paolo Xausa, Oct 17 2024 *)

a(n) = 2*3^(n-2) + 3^floor((n-2)/2).
From Stefano Spezia, Jul 20 2024: (Start)
G.f.: x*(1 - 5*x^2)/((1 - 3*x)*(1 - 3*x^2)).
E.g.f.: (2*cosh(3*x) + 3*cosh(sqrt(3)*x) + 2*sinh(3*x) + sqrt(3)*sinh(sqrt(3)*x) - 5)/9. (End)

A329910 Number of harmoniously labeled graphs with n edges and at most n vertices.

Original entry on oeis.org

0, 0, 1, 4, 32, 72, 2187, 20736, 262144, 3200000, 48828125, 729000000, 13060694016, 230539333248, 4747561509943, 96717311574016, 2251799813685250, 51998697814229000, 1350851717672990000, 34867844010000000000, 1000000000000000000000, 28531167061100000000000

Offset: 1

Christian Barrientos, Nov 23 2019

A graph G with n edges is harmonious if there is an injection f from its vertex set to the group of integers modulo n such that when each edge uv of G is assigned the weight f(u)+f(v) (mod n), the resulting weights are distinct.

			a(3)=1 because there is only one harmonious graph with 3 edges and at most 3 vertices.

R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404.
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs

A085526 contains the odd-indexed terms.

Table[If[EvenQ[n],(n*(n-2)/4)^(n/2),((n-1)/2)^n],{n,1,22}] (* Stefano Spezia, Nov 24 2019 *)

For n odd, a(n) = ((n-1)/2)^n. For n even, a(n) = (n*(n-2)/4)^(n/2).

A308203 Array read by ascending antidiagonals: T(n,k) = number of non-isomorphic kC_n-snakes for n >= 3 and k >= 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 3, 6, 1, 1, 3, 6, 6, 10, 1, 1, 4, 6, 18, 10, 20, 1, 1, 4, 10, 18, 45, 20, 36, 1, 1, 5, 10, 40, 45, 135, 36, 72, 1, 1, 5, 15, 40, 136, 135, 378, 72, 136, 1, 1, 6, 15, 75, 136, 544, 378, 1134, 136, 272, 1

Offset: 3

Christian Barrientos, May 15 2019

A kC_n-snake is a connected graph in which the k >= 2 blocks are isomorphic to the cycle C_n and the block-cutpoint graph is a path.

			T(n,2)=1 because there is only one way to connect two copies of C_n.
T(3,k)=1 because C_3 is isomorphic to K_3 and all the selections of 2 cutpoints, in each interior copy of C_3, are equivalent.
T(5,4)=3 there are only 3 non-equivalent strings of length 2 corresponding to the distances between consecutive cutpoints: 11, 12, and 2,2.
Table begins:
1    1    1     1      1       1        1         1          1           1            1
1    2    3     6     10      20       36        72        136         272          528
1    2    3     6     10      20       36        72        136         272          528
1    3    6    18     45     135      378      1134       3321        9963        29646
1    3    6    18     45     135      378      1134       3321        9963        29646
1    4   10    40    136     544     2080      8320      32896      131584       524800
1    4   10    40    136     544     2080      8320      32896      131584       524800
1    5   15    75    325    1625     7875     39375     195625      978125      4884375
1    5   15    75    325    1625     7875     39375     195625      978125      4884375
1    6   21   126    666    3996    23436    140616     840456     5042736     30236976
1    6   21   126    666    3996    23436    140616     840456     5042736     30236976
1    7   28   196   1225    8575    58996    412972    2883601    20185207    141246028
1    7   28   196   1225    8575    58996    412972    2883601    20185207    141246028
1    8   36   288   2080   16640   131328   1050624    8390656    67125248    536887296
1    8   36   288   2080   16640   131328   1050624    8390656    67125248    536887296
1    9   45   405   3321   29889   266085   2394765   21526641   193739769   1743421725
1    9   45   405   3321   29889   266085   2394765   21526641   193739769   1743421725
1   10   55   550   5050   50500   500500   5005000   50005000   500050000   5000050000

For n >= 3 and k >= 2, T(n,k) = (floor(n/2)^(k-2) + floor(n/2)^(floor(k-1)/2))/2.

For n even, T(n, 2)=1, if k is odd T(n,k)=(n/2)*T(n,k-1), if k is even T(n,k)=(n/2)*T(n,k-1)-((n-2)/4)*(n/2)^((k-2)/2).

A317489 Irregular triangle read by rows. For n >= 3 and 1 <= k <= floor(n/3), T(n,k) is the number of palindromic compositions of n into k parts of size at least 3.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 0, 3, 0, 1, 1, 3, 2, 1, 0, 4, 0, 1, 1, 1, 4, 3, 1, 1, 0, 5, 0, 3, 1, 1, 5, 4, 3, 1, 1, 0, 6, 0, 6, 0, 1, 1, 6, 5, 6, 3, 1, 0, 7, 0, 10, 0, 1, 1, 1, 7, 6, 10, 6, 1, 1, 0, 8, 0, 15, 0, 4, 1, 1, 8, 7, 15, 10, 4, 1, 1, 0, 9, 0, 21, 0, 10, 0, 1, 1, 9, 8, 21, 15, 10, 4, 1, 0, 10, 0, 28, 0, 20, 0, 1, 1, 1, 10, 9, 28, 21, 20, 10, 1, 1, 0, 11, 0, 36, 0, 35, 0, 5

Offset: 3

Christian Barrientos and Sarah Minion, Jul 29 2018

			For n=24 and k=3, T(24,3) = 8 = binomial((24-2)/2-3, (3-1)/2) = binomial(8,1).
The first entries of the irregular triangle formed by the values of T(n,k) are:
  1;
  1;
  1;
  1,  1;
  1,  0;
  1,  1;
  1,  0,  1;
  1,  1,  1;
  1,  0,  2;
  1,  1,  2,  1;
  1,  0,  3,  0;
  1,  1,  3,  2;
  1,  0,  4,  0,  1;
  1,  1,  4,  3,  1;
  1,  0,  5,  0,  3;
  1,  1,  5,  4,  3,  1;
  1,  0,  6,  0,  6,  0;
  1,  1,  6,  5,  6,  3;
  1,  0,  7,  0, 10,  0,  1;
  1,  1,  7,  6, 10,  6,  1;
  1,  0,  8,  0, 15,  0,  4;
  1,  1,  8,  7, 15, 10,  4,  1;
  1,  0,  9,  0, 21,  0, 10,  0;
  1,  1,  9,  8, 21, 15, 10,  4;
  1,  0, 10,  0, 28,  0, 20,  0,  1;
  1,  1, 10,  9, 28, 21, 20, 10,  1;
  1,  0, 11,  0, 36,  0, 35,  0,  5;

Row sums of the triangle equal A226916(n+4).

T[n_, k_] := If[Mod[n, 2] == 1 && Mod[k, 2] == 0, 0, Binomial[Quotient[n-1, 2] - k, Quotient[k-1, 2]]];
Table[T[n, k], {n, 3, 30}, {k, 1, Quotient[n, 3]}] // Flatten (* Jean-François Alcover, Sep 13 2018, from PARI *)

T(n,k)=if(n%2==1&&k%2==0, 0,  binomial((n-1)\2-k, (k-1)\2)); \\ Andrew Howroyd, Sep 07 2018

T(n,k) = 0 if n is odd and k is even;
T(n,k) = binomial((n-1)/2-k,(k-1)/2) if n is odd and k is odd;
T(n,k) = binomial((n-2)/2-k,(k-1)/2) if n is even and k is odd;
T(n,k) = binomial((n-2)/2-k,(k-2)/2) if n is even and k is even.

A255908 Triangle read by rows: T(n,L) = number of rho-labeled graphs with n edges whose labeling is bipartite with boundary value L.

Original entry on oeis.org

2, 4, 8, 8, 32, 48, 16, 128, 288, 384, 32, 512, 1728, 3072, 3840, 64, 2048, 10368, 24576, 38400, 46080, 128, 8192, 62208, 196608, 384000, 552960, 645120, 256, 32768, 373248, 1572864, 3840000, 6635520, 9031680, 10321920, 512, 131072, 2239488, 12582912, 38400000, 79626240, 126443520, 165150720, 185794560, 1024, 524288, 13436928, 100663296, 384000000, 955514880, 1770209280, 2642411520, 3344302080, 3715891200

Offset: 1

Christian Barrientos and Sarah Minion, Mar 10 2015

A graph with n edges is rho-labeled if there exists a one-to-one mapping from its vertex set to {0,1,...,2n} such that every edge receives as label the absolute difference of its end-vertices and the edge labels are x_1, x_2, ..., x_n where x_i = i or x_i = 2n + 1 - i. A rho-labeling of a bipartite graph is said to be bipartite when the labels of one stable set are smaller than the labels on the other stable set. The largest of the smaller vertex labels is its boundary value.
From Robert G. Wilson v, Jul 05 2015: (Start)
The columns:
T(n, 0) = 2^n,
T(n, 1) = 2^(2n-1),
T(n, 2) = 2^(n+1)*3^(n-2),
T(n, 3) = 3*2^(3n-5),
T(n, 4) = 3*2^(n+3)*5^(n-4),
T(n, 5) = 5*2^(2n-2)*3^(n-4), etc.
The diagonals:
the main,            T(n, n-1) = 2^n*n*(n-1!) = 2*A002866,
the second diagonal, T(n, n-2) = 2^n*(n-1)^2*(n-2)! = 4*A014479,
the third diagonal,  T(n, n-3) = 2^n*(n-2)^3*(n-3)!,
the k_th diagonal,   T(n, n-k) = 2^n*(n-k)^k*(n-k)!, etc.
... (End)

			For n=5 and L=1, T(5,1)=(2^5)*(1!)*(1+1)^(5-1)=512.
For n=9 and L=3, T(9,3)=12582912.
Triangle, T, begins:
-----------------------------------------------------------------------------
n\L |   0       1         2          3          4          5           6
----|------------------------------------------------------------------------
1   |   2;
2   |   4,      8;
3   |   8,     32,       48;
4   |  16,    128,      288,       384;
5   |  32,    512,     1728,      3072,      3840;
6   |  64,   2048,    10368,     24576,     38400,     46080;
7   | 128,   8192,    62208,    196608,    384000,    552960,     645120;
8   | 256,  32768,   373248,   1572864,   3840000,   6635520,    9031680, ...
...
For n=2 and L=1, T(2,1)=8, because: the bipartite graph <{v1,v2,v3},{x1=v1v2,x2=v1v3}> has rho-labelings (2,1,3),(2,1,4) with L=1 on the stable set {x2} and rho-labelings (1,2,0),(0,4,1) with L=1 on the stable set {x1,x3}; the bipartite graph <{v1,v2,v3,v4},{x1=v1v2,x2=v3v4}> has rho-labeling (0,4,1,3),(1,2,0,3) with L=1 on the stable set {v1,v3} and rho-labeling (4,0,3,1),(2,1,3,0) with L=1 on the stable set {v2,v4}. - _Danny Rorabaugh_, Apr 03 2015

Joseph A. Gallian, A dynamic survey of graph labeling, Elec. J. Combin., (2014), #DS6.

[2^n*Factorial(l)*(l+1)^(n-l): l in [0..n-1], n in [1..10]]; // Bruno Berselli, Aug 05 2015

t[n_, l_] := 2^n*l!(l+1)^(n-l); Table[ t[n, l], {n, 8}, {l, 0, n-1}] // Flatten (* Robert G. Wilson v, Jul 05 2015 *)

For n>=1, 0<=L<=n-1, T(n,L)=(2^n)*(L!)*(L+1)^(n-L).

A245519 Number of alpha-labeled graphs with n edges and at most n vertices.

Original entry on oeis.org

0, 0, 0, 2, 10, 64, 336, 1872, 11104, 71944, 508032, 3511232, 27192704, 223750464, 1947253504, 17899536448, 173156535168, 1760383827776, 18752453106176, 209034916385472, 2432351796434560, 29509268795249700

Offset: 1

Sarah Minion and Christian Barrientos, Jul 24 2014

			For n=4, a(4)=2, there are 2 alpha-labeled graphs with 4 edges and at most 4 vertices.
For n=10, a(10)=71944, there are 71944 alpha-labeled graphs with 10 edges and at most 10 vertices.

Christian Barrientos, Sarah Minion, On the number of alpha-labeled graphs, Discussiones Mathematicae Graph Theory, to appear.
J. A. Gallian, A dynamic survey of graph labeling, Elec. J. Combin., (2013), #DS6.
David A. Sheppard, The factorial representation of major balanced labelled graphs, Discrete Math., 15(1976), no. 4, 379-388.

Cf. A241094, A005193.

a(n) = Sum_{L=1..n-2} Sum_{i=1..n-1} Product_{k=1..n} d(L,k,i), where
for i < L,
            d(L,k)       if 1 <= k <= i,
d(L,k,i) ={ d(L,k) - 1   if i < k < n - i,
            d(L,k)       if n - i <= k <= n;
for i > L + 1,
            d(L,k)       if 1 <= k <= n - i,
d(L,k,i) ={ d(L,k) - 1   if n - i < k < n - i + L + 2,
            d(L,k)       if n - i + L + 2 <= k <= n.
          k          if 1 <= k < m,
d(L,k) ={ L + 1      if m <= k <= M,
          n + 1 - k  if M < k <= n,
m = min{L + 1, n - L}, M = max{L + 1, n - L}.

A245518 Irregular triangle read by rows: T(n,i) = number of alpha-labeled graphs with n edges that do not use the label i, for 1 <= i <= n-1 and n >= 4.

Original entry on oeis.org

1, 0, 1, 4, 2, 2, 4, 16, 12, 8, 12, 16, 64, 64, 40, 40, 64, 64, 284, 328, 236, 176, 236, 328, 284, 1360, 1760, 1432, 1000, 1000, 1432, 1760, 1360, 7184, 9928, 9092, 6536, 5312, 6536, 9092, 9928, 7184

Offset: 4

Sarah Minion and Christian Barrientos, Jul 24 2014

			For n=4 and i=2, a(4,2) = 0.
For n=8 and i=5, a(8,5) = 64.
Triangle begins:
[n\i] [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]     [9]
[4]    1,      0,      1;
[5]    4,      2,      2,      4;
[6]    16,     12,     8,      12,     16;
[7]    64,     64,     40,     40,     64,     64;
[8]    284,    328,    236,    176,    236,    328,    284;
[9]    1360,   1760,   1432,   1000,   1000,   1432,   1760,   1360;
[10]   7184,   9928,   9092,   6536,   5312,   6536,   9092,   9928,   7184;
. . .

Christian Barrientos, Sarah Minion, On the number of alpha-labeled graphs, Discussiones Mathematicae Graph Theory, to appear.
J. A. Gallian, A dynamic survey of graph labeling, Elec. J. Combin., (2013), #DS6.
David A. Sheppard, The factorial representation of major balanced labelled graphs, Discrete Math., 15(1976), no. 4, 379-388.

Cf. A241094, A005193.

a(n,i) = sum_{L=1..^n-2} product_{k=1..n} d(L,k,i), where
for i < L,
            d(L,k)       if 1 <= k <= i,
d(L,k,i) ={ d(L,k) - 1   if i < k < n - i,
            d(L,k)       if n - i <= k <= n;
for i > L + 1,
            d(L,k)       if 1 <= k <= n - i,
d(L,k,i) ={ d(L,k) - 1   if n - i < k < n - i + L + 2,
            d(L,k)       if n - i + L + 2 <= k <= n.
          k          if 1 <= k < m,
d(L,k) ={ L + 1      if m <= k <= M,
          n + 1 - k  if M < k <= n,
m = min{L + 1, n - L}, M = max{L + 1, n - L}.

A245517 Irregular triangle read by rows: T(n,L) = number of alpha-labeled graphs with n edges and boundary value L that do not use one number from (1,2,...,n-1) as a label (n >= 4, 1 <= L <= n - 2).

Original entry on oeis.org

1, 1, 4, 4, 4, 12, 20, 20, 12, 32, 88, 96, 88, 32, 80, 352, 504, 504, 352, 80, 192, 1328, 2592, 2880, 2592, 1328, 192, 448, 4816, 12852, 17280, 17280, 12852, 4816, 448

Offset: 4

Sarah Minion and Christian Barrientos, Jul 24 2014

			For n=9 and L=5, T(9,5) = 2592.
For n=10 and L=4, T(10,4) = 17280.
Triangle begins:
[n\L]  [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]
[4]     1,      1;
[5]     4,      4,      4;
[6]     12,     20,     20,     12;
[7]     32,     88,     96,     88,     32;
[8]     80,     352,    504,    504,    352,    80;
[9]     192,    1328,   2592,   2880,   2592,   1328,   192;
[10]    448,    4816,   12852,  17280,  17280,  12852,  4816,   448;
...

Christian Barrientos, Sarah Minion, On the number of alpha-labeled graphs, Discussiones Mathematicae Graph Theory, to appear.
J. A. Gallian, A dynamic survey of graph labeling, Elec. J. Combin., (2013), #DS6.
David A. Sheppard, The factorial representation of major balanced labelled graphs, Discrete Math., 15(1976), no. 4, 379-388.

Cf. A241094, A005193.

a(n,L,i) = \sum_{i = 1}^{n - 1} \prod_{k = 1}^{n} d(L,k,i), where
for i < L,
            d(L,k)       if 1 <= k <= i,
d(L,k,i) ={ d(L,k) - 1   if i < k < n - i,
            d(L,k)       if n - i <= k <= n;
for i > L + 1,
            d(L,k)       if 1 <= k <= n - i,
d(L,k,i) ={ d(L,k) - 1   if n - i < k < n - i + L + 2,
            d(L,k)       if n - i + L + 2 <= k <= n.
          k          if 1 <= k < m,
d(L,k) ={ L + 1      if m <= k <= M,
          n + 1 - k  if M < k <= n,
m = min{L + 1, n - L}, M = max{L + 1, n - L}.

A241094 Triangle read by rows: T(n,i) = number of gracefully labeled graphs with n edges that do not use the label i, 1 <= i <= n-1, n > 1.

Original entry on oeis.org

0, 1, 1, 4, 4, 4, 18, 24, 24, 18, 96, 144, 144, 96, 600, 960, 1080, 1080, 960, 600, 4320, 7200, 8460, 8460, 8460, 7200, 4320, 35280, 60840, 75600, 80640, 80640, 75600, 60480, 35280, 322560, 564480, 725760, 806400, 806400, 806400, 725760, 564480, 322560

Offset: 2

Christian Barrientos and Sarah Minion, Apr 15 2014

A graph with n edges is graceful if its vertices can be labeled with distinct integers in the range 0,1,...,n in such a way that when the edges are labeled with the absolute differences between the labels of their end-vertices, the n edges have the distinct labels 1,2,...,n.

			For n=7 and i=3, g(7,3) = 1080.
For n=7 and i=5, g(7,5) = 960.
Triangle begins:
[n\i]  [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]
[2]     0;
[3]     1,      1;
[4]     4,      4,      4;
[5]    18,     24,     24,     18;
[6]    96,    144,    144,    144,     96;
[7]   600,    960,   1080,   1080,    960,    600;
[8]  4320,   7200,   8640,   8640,   8640,   7200,   4320;
[9] 35280,  60480,  75600,  80640,  80640,  75600,  60480,  35280;
...
- _Bruno Berselli_, Apr 23 2014

C. Barrientos and S. M. Minion, Enumerating families of labeled graphs, J. Integer Seq., 18(2015), article 15.1.7.
J. A. Gallian, A dynamic survey of graph labeling, Elec. J. Combin., (2013), #DS6.
David A. Sheppard, The factorial representation of major balanced labelled graphs, Discrete Math., 15(1976), no. 4, 379-388.

Cf. A001563, A003022, A004137, A005488, A006967, A033472, A081621, A103300, A117747, A212661.

/* As triangle: */ [[i le Floor(n/2) select Factorial(n-2)*(n-1-i)*i else Factorial(n-2)*(n-i)*(i-1): i in [1..n-1]]: n in [2..10]]; // Bruno Berselli, Apr 23 2014

Labeled:=(i,n) piecewise(n<2 or i<1, -infinity, 1 <= i <= floor(n/2), GAMMA(n-1)*(n-1-i)*i, ceil((n+1)/2) <= i <= n-1, GAMMA(n-1)*(n-i)*(i-1), infinity):

n=10; (* This number must be replaced every time in order to produce the different entries of the sequence *)
For[i = 1, i <= Floor[n/2], i++, g[n_,i_]:=(n-2)!*(n-1-i)*i; Print["g(",n,",",i,")=", g[n,i]]]
For[i = Ceiling[(n+1)/2], i <= (n-1), i++, g[n_,i_]:=(n-2)!*(n-i)*(i-1); Print["g(",n,",",i,")=",g[n,i]]]

For n >=2, if 1 <= i <= floor(n/2), g(n,i) = (n-2)!*(n-1-i)*i; if ceiling((n+1)/2) <= i <= n-1, g(n,i) = (n-2)!*(n-i)*(i-1).
# alternative
A241094 := proc(n,i)
    if n <2 or i<1 or i >= n then
        0;
    elif i <= floor(n/2) then
        GAMMA(n-1)*(n-1-i)*i;
    else
        GAMMA(n-1)*(n-i)*(i-1) ;
    fi ;
end proc:
seq(seq(A241094(n,i),i=1..n-1),n=2..12); # R. J. Mathar, Jul 30 2024

cp's OEIS Frontend

User: Christian Barrientos

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A329910 Number of harmoniously labeled graphs with n edges and at most n vertices.

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A308203 Array read by ascending antidiagonals: T(n,k) = number of non-isomorphic kC_n-snakes for n >= 3 and k >= 2.

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A317489 Irregular triangle read by rows. For n >= 3 and 1 <= k <= floor(n/3), T(n,k) is the number of palindromic compositions of n into k parts of size at least 3.

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A255908 Triangle read by rows: T(n,L) = number of rho-labeled graphs with n edges whose labeling is bipartite with boundary value L.

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A245519 Number of alpha-labeled graphs with n edges and at most n vertices.

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