A348815 -id:A348815 - cp's OEIS frontend

A349812 Triangle read by rows: row 1 is [1]; for n >= 1, row n gives coefficients of expansion of (-1/x + x)*(1/x + 1 + x)^(n-1) in order of increasing powers of x.

1, -1, 0, 1, -1, -1, 0, 1, 1, -1, -2, -2, 0, 2, 2, 1, -1, -3, -5, -4, 0, 4, 5, 3, 1, -1, -4, -9, -12, -9, 0, 9, 12, 9, 4, 1, -1, -5, -14, -25, -30, -21, 0, 21, 30, 25, 14, 5, 1, -1, -6, -20, -44, -69, -76, -51, 0, 51, 76, 69, 44, 20, 6, 1, -1, -7, -27, -70, -133, -189, -196, -127, 0, 127, 196, 189, 133, 70, 27, 7, 1

Offset: 0

N. J. A. Sloane, Dec 23 2021

The rule for constructing this triangle (ignoring row 0) is the same as that for A027907: each number is the sum of the three numbers immediately above it in the previous row. Here row 1 is [-1, 0, 1] instead of [1, 1, 1].

			Triangle begins:
   1;
  -1,  0,   1;
  -1, -1,   0,   1,    1;
  -1, -2,  -2,   0,    2,    2,    1;
  -1, -3,  -5,  -4,    0,    4,    5,    3,  1;
  -1, -4,  -9, -12,   -9,    0,    9,   12,  9,   4,   1;
  -1, -5, -14, -25,  -30,  -21,    0,   21, 30,  25,  14,   5,   1;
  -1, -6, -20, -44,  -69,  -76,  -51,    0, 51,  76,  69,  44,  20,  6,  1;
  -1, -7, -27, -70, -133, -189, -196, -127,  0, 127, 196, 189, 133, 70, 27, 7, 1;
  ...

Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965 [Mentions application to design of antenna arrays. Annotated scan.]

Cf. A007318, A027907, A112467, A349813, A348815.

The left half of the triangle is A026300, the right half is A064189 (or A122896). The central (nonzero) column gives the Motzkin numbers A001006.

t1:=-1/x+x; m:=1/x+1+x;
lprint([1]);
for n from 1 to 12 do
w1:=expand(t1*m^(n-1));
w3:=expand(x^n*w1);
w4:=series(w3,x,2*n+1);
w5:=seriestolist(w4);
lprint(w5);
od:

A369923 Array read by antidiagonals: A(n,k) is the number of permutations of n copies of 1..k with values introduced in order and without cyclically adjacent elements equal.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 31, 22, 1, 0, 1, 293, 1415, 134, 1, 0, 1, 3326, 140343, 75843, 866, 1, 0, 1, 44189, 20167651, 83002866, 4446741, 5812, 1, 0, 1, 673471, 3980871156, 158861646466, 55279816356, 276154969, 40048, 1, 0

Offset: 1

Andrew Howroyd, Feb 05 2024

Also, T(n,k) is the number of generalized chord labeled loopless diagrams with k parts of K_n. See the Krasko reference for a full definition.

			Array begins:
n\k| 1 2    3         4              5                    6 ...
---+-----------------------------------------------------------
 1 | 0 1    1         1              1                    1 ...
 2 | 0 1    4        31            293                 3326 ...
 3 | 0 1   22      1415         140343             20167651 ...
 4 | 0 1  134     75843       83002866         158861646466 ...
 5 | 0 1  866   4446741    55279816356     1450728060971387 ...
 6 | 0 1 5812 276154969 39738077935264 14571371516350429940 ...
 ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 51 antidiagonals)
Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017.
Mathematics.StackExchange, Find the number of k 1's, k 2's, ... , k n's - total kn cards, Apr 08 2012.

Rows 2..6 are A003436, A348813, A348815, A348818, A348821.
Column 3 is A197657, column 4 appears to be A209183(n)/2.
Cf. A322013 (without linearly adjacent elements equal), A322093.

T[n_, k_] := If[k == 1, 0, Expand[(-1)^(k (n + 1))/(k - 1)! n Hypergeometric1F1[1 - n, 2, x]^k x^(k - 1)] /. x^p_ :> p!] (* Eric W. Weisstein, Feb 20 2025 *)

\\ compare with A322013.
q(n, x) = sum(i=1, n, (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!)
T(n, k) = if(k > 1, subst(serlaplace(n*q(n, x)^k/x), x, 1)/(k-1)!, 0)

A348813 a(n) = number of chord labeled loopless diagrams by number of K_3.

Original entry on oeis.org

0, 1, 22, 1415, 140343, 20167651, 3980871156, 1035707510307, 343866839138005, 141979144588872613, 71386289535825383386, 42954342000612934599071, 30482693813120122213093587, 25196997894058490607106028095, 24001522306527907199721466108488, 26102037346800387738363882455862531

Offset: 1

Michael De Vlieger, Nov 01 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017. See Appendix Table 2.

Row 3 of A369923.

Cf. A190826, A348814, A292409; A003436, A348815, A348818, A348821.

a(14) onwards from Andrew Howroyd, Feb 05 2024

A348818 a(n) = number of chord labeled loopless diagrams by number of K_5.

Original entry on oeis.org

0, 1, 866, 4446741, 55279816356, 1450728060971387, 72078730629785795963, 6235048155225093080061949, 879601407931825739964190440635, 192100729970218737700046212217095291, 62258393664270652226502315136978421947948, 28913744296806659870889046765907226809528931041

Offset: 1

Michael De Vlieger, Nov 01 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100
Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017. See Appendix Table 4.

Row 5 of A369923.

Cf. A190833, A348819, A348820; A003436, A348813, A348815, A348821.

a(9) onwards from Andrew Howroyd, Feb 05 2024

A348821 a(n) = number of chord labeled loopless diagrams by number of K_6.

Original entry on oeis.org

0, 1, 5812, 276154969, 39738077935264, 14571371516350429940, 11876790400066163254723167, 19372051918038657958659363247949, 58256941603805590330534264712744407687, 302616041649108508974263266688425815263488561, 2575195630881373033515248134269171034879932771154311

Offset: 1

Michael De Vlieger, Nov 01 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100
Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017. See Appendix Table 5.

Row 6 of A369923.

Cf. A190835, A348822, A348823; A003436, A348813, A348815, A348818.

a(8) onwards from Andrew Howroyd, Feb 05 2024

A348816 a(n) = number of loopless diagrams by number of K_4 up to rotational symmetry.

Original entry on oeis.org

0, 1, 15, 4790, 4151415, 6619291247, 17510518983528, 71631394311300461, 429426878302882412435, 3616596939726424941979785

Offset: 1

Michael De Vlieger, Nov 01 2021

Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017. See Appendix Table 3.

Cf. A190830, A348815, A348817; A007474, A348814, A348819, A348822.

A348817 a(n) = number of loopless diagrams by number of K_4 up to all symmetries.

Original entry on oeis.org

0, 1, 13, 2576, 2081393, 3309962320, 8755277273334, 35815698613833466, 214713439275724149414, 1808298469877117320495867

Offset: 1

Michael De Vlieger, Nov 01 2021

Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017. See Appendix Table 3.

Cf. A190830, A348815, A348816; A003437, A292409, A348820, A348823.

A349816 Irregular triangle read by rows: the right-hand side of the triangle in A349815.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 4, 4, 3, 1, 8, 13, 12, 8, 4, 1, 13, 33, 41, 37, 25, 13, 5, 1, 74, 124, 136, 116, 80, 44, 19, 6, 1, 124, 334, 450, 456, 376, 259, 149, 70, 26, 7, 1, 784, 1364, 1616, 1541, 1240, 854, 504, 252, 104, 34, 8, 1, 1364, 3764, 5305, 5761, 5251, 4139, 2850, 1714, 894, 398, 147, 43, 9, 1, 9069, 16194, 20081, 20456, 18001, 13954, 9597, 5856, 3153, 1482, 597, 200, 53, 10, 1

Offset: 0

N. J. A. Sloane, Dec 23 2021

It seems more symmetrical to omit the central column of zeros in A349815.

			Triangle begins:
    1;
    1,    1;
    1,    2,    1;
    2,    4,    4,    3,    1;
    8,   13,   12,    8,    4,   1;
   13,   33,   41,   37,   25,  13,   5,   1;
   74,  124,  136,  116,   80,  44,  19,   6,   1;
  124,  334,  450,  456,  376, 259, 149,  70,  26,  7, 1;
  784, 1364, 1616, 1541, 1240, 854, 504, 252, 104, 34, 8, 1;
  ...

Cf. A348815, A349818.

cp's OEIS Frontend

A349812 Triangle read by rows: row 1 is [1]; for n >= 1, row n gives coefficients of expansion of (-1/x + x)*(1/x + 1 + x)^(n-1) in order of increasing powers of x.

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A369923 Array read by antidiagonals: A(n,k) is the number of permutations of n copies of 1..k with values introduced in order and without cyclically adjacent elements equal.

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A348813 a(n) = number of chord labeled loopless diagrams by number of K_3.

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A348818 a(n) = number of chord labeled loopless diagrams by number of K_5.

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A348821 a(n) = number of chord labeled loopless diagrams by number of K_6.

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A348816 a(n) = number of loopless diagrams by number of K_4 up to rotational symmetry.

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A348817 a(n) = number of loopless diagrams by number of K_4 up to all symmetries.

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A349816 Irregular triangle read by rows: the right-hand side of the triangle in A349815.

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