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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.

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: Sarah Minion

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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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.

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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).

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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.

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