A120908 -id:A120908 - cp's OEIS frontend

A269276 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally or antidiagonally adjacent neighbor totalling three exactly once.

0, 4, 0, 24, 108, 0, 108, 1368, 1620, 0, 432, 13896, 46872, 20412, 0, 1620, 127512, 1104264, 1365336, 236196, 0, 5832, 1104264, 23549400, 74853576, 36673560, 2598156, 0, 20412, 9211608, 474819408, 3719884392, 4684312584, 938176344

Offset: 1

R. H. Hardin, Feb 21 2016

Table starts
.0........4..........24............108...............432................1620
.0......108........1368..........13896............127512.............1104264
.0.....1620.......46872........1104264..........23549400...........474819408
.0....20412.....1365336.......74853576........3719884392........174924572760
.0...236196....36673560.....4684312584......542973139128......59587625651904
.0..2598156...938176344...279339197256....75556007986536...19356924219624936
.0.27634932.23230366488.16128206816904.10181956012212600.6090616046325570480

			Some solutions for n=3 k=4
..0..2..0..1. .0..0..0..0. .0..2..0..0. .0..0..0..0. .0..0..2..0
..0..2..3..1. .2..2..0..0. .2..2..3..1. .2..0..2..1. .0..2..2..3
..2..1..0..2. .3..2..1..1. .0..1..0..2. .0..0..1..3. .3..1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..241

Row 1 is A120908.

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 18*a(n-1) -81*a(n-2)
k=3: a(n) = 42*a(n-1) -441*a(n-2)
k=4: a(n) = 98*a(n-1) -2401*a(n-2) for n>3
k=5: a(n) = 234*a(n-1) -14277*a(n-2) +68796*a(n-3) -86436*a(n-4)
k=6: [order 6] for n>7
k=7: [order 10] for n>11
Empirical for row n:
n=1: a(n) = 6*a(n-1) -9*a(n-2)
n=2: a(n) = 14*a(n-1) -49*a(n-2) for n>4
n=3: a(n) = 36*a(n-1) -378*a(n-2) +972*a(n-3) -729*a(n-4) for n>7
n=4: [order 8] for n>12
n=5: [order 18] for n>23
n=6: [order 40] for n>46

A269214 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally, diagonally or antidiagonally adjacent neighbor totalling three exactly once.

Original entry on oeis.org

0, 4, 0, 24, 96, 0, 108, 768, 1152, 0, 432, 6528, 18048, 11424, 0, 1620, 49536, 308544, 361728, 103488, 0, 5832, 360960, 4744704, 12548544, 6712704, 889056, 0, 20412, 2546304, 70371048, 394072704, 474091776, 118872576, 7375872, 0, 69984, 17563392

Offset: 1

R. H. Hardin, Feb 20 2016

Table starts
.0........4..........24............108...............432................1620
.0.......96.........768...........6528.............49536..............360960
.0.....1152.......18048.........308544...........4744704............70371048
.0....11424......361728.......12548544.........394072704.........11985002256
.0...103488.....6712704......474091776.......30541426560.......1910809190712
.0...889056...118872576....17118725376.....2267772823680.....292321215814512
.0..7375872..2039727744...599456856000...163535201141376...43468685827935816
.0.59698464.34214296320.20531285093184.11544796423498368.6331185189881558208

			Some solutions for n=3 k=4
..2..2..3..2. .2..2..2..3. .0..1..0..2. .0..0..0..2. .2..2..0..0
..0..2..0..2. .0..0..1..0. .1..1..3..0. .0..1..2..0. .0..0..2..3
..2..1..0..1. .0..1..0..1. .0..1..0..2. .0..0..0..0. .0..0..1..3

R. H. Hardin, Table of n, a(n) for n = 1..161

Column 2 is A269091.

Row 1 is A120908.

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 14*a(n-1) -49*a(n-2) for n>3
k=3: a(n) = 30*a(n-1) -237*a(n-2) +180*a(n-3) -36*a(n-4) for n>5
k=4: [order 6] for n>7
k=5: [order 20] for n>21
k=6: [order 42] for n>43
Empirical for row n:
n=1: a(n) = 6*a(n-1) -9*a(n-2)
n=2: a(n) = 10*a(n-1) -13*a(n-2) -60*a(n-3) -36*a(n-4)
n=3: [order 8]
n=4: [order 20]
n=5: [order 52] for n>53

A269097 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally or vertically adjacent neighbor totalling three exactly once.

Original entry on oeis.org

0, 4, 4, 24, 96, 24, 108, 1152, 1152, 108, 432, 11424, 31296, 11424, 432, 1620, 103488, 715320, 715320, 103488, 1620, 5832, 889056, 15024096, 37963968, 15024096, 889056, 5832, 20412, 7375872, 300056400, 1856325000, 1856325000, 300056400, 7375872

Offset: 1

R. H. Hardin, Feb 19 2016

Table starts
......0..........4.............24................108....................432
......4.........96...........1152..............11424.................103488
.....24.......1152..........31296.............715320...............15024096
....108......11424.........715320...........37963968.............1856325000
....432.....103488.......15024096.........1856325000...........211625837280
...1620.....889056......300056400........86415197088.........22984646281176
...5832....7375872.....5795398368......3892946216856.......2416707037884480
..20412...59698464...109294975080....171307237216320.....248267055360211392
..69984..474360768..2024660774592...7406621052177000...25062609875880382656
.236196.3715826016.36988673403168.315868041326151072.2495854741014485439624

			Some solutions for n=3 k=4
..0..1..3..2. .0..0..2..2. .2..2..2..3. .0..2..2..0. .0..0..1..1
..0..3..1..0. .0..2..3..3. .0..2..2..2. .0..2..0..0. .0..2..0..0
..0..2..3..2. .1..1..1..3. .2..0..0..3. .0..2..3..2. .2..2..1..1

R. H. Hardin, Table of n, a(n) for n = 1..264

Column 1 is A120908.

Empirical for column k:
k=1: a(n) = 6*a(n-1) -9*a(n-2)
k=2: a(n) = 14*a(n-1) -49*a(n-2) for n>3
k=3: a(n) = 36*a(n-1) -378*a(n-2) +972*a(n-3) -729*a(n-4) for n>5
k=4: [order 6] for n>7
k=5: [order 14] for n>15
k=6: [order 26] for n>27
k=7: [order 64] for n>65

A269152 T(n,k) = Number of n X k 0..3 arrays with some element plus some horizontally, antidiagonally or vertically adjacent neighbor totalling three exactly once.

Original entry on oeis.org

0, 4, 4, 24, 80, 24, 108, 768, 768, 108, 432, 6224, 13904, 6224, 432, 1620, 46464, 220968, 220968, 46464, 1620, 5832, 330192, 3277728, 7002040, 3277728, 330192, 5832, 20412, 2270592, 46576336, 208984848, 208984848, 46576336, 2270592, 20412

Offset: 1

R. H. Hardin, Feb 20 2016

Table starts
......0.........4............24..............108.................432
......4........80...........768.............6224...............46464
.....24.......768.........13904...........220968.............3277728
....108......6224........220968..........7002040...........208984848
....432.....46464.......3277728........208984848.........12637025328
...1620....330192......46576336.......6004186984........738478504448
...5832...2270592.....642676704.....167970539096......42119837369168
..20412..15251152....8680278136....4607603633440....2359047063894464
..69984.100647168..115349343264..124496158984840..130272136732736736
.236196.655139152.1513379596864.3323815506994632.7113223023541150960

			Some solutions for n=3, k=4
..2..2..3..1. .0..2..2..2. .2..0..1..1. .0..3..3..1. .2..2..3..3
..3..2..3..3. .3..3..3..2. .2..0..1..0. .1..1..3..1. .3..3..1..2
..0..2..3..3. .3..2..2..2. .1..0..0..2. .3..3..3..1. .2..3..3..3

R. H. Hardin, Table of n, a(n) for n = 1..220

Column 1 is A120908.

Empirical for column k:
k=1: a(n) = 6*a(n-1) -9*a(n-2)
k=2: a(n) = 12*a(n-1) -38*a(n-2) +12*a(n-3) -a(n-4) for n>5
k=3: [order 8] for n>9
k=4: [order 20] for n>22
k=5: [order 42] for n>45

A269186 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally, diagonally, antidiagonally or vertically adjacent neighbor totalling three exactly once.

Original entry on oeis.org

0, 4, 4, 24, 48, 24, 108, 384, 384, 108, 432, 2736, 5888, 2736, 432, 1620, 18336, 80112, 80112, 18336, 1620, 5832, 118032, 1031344, 2097552, 1031344, 118032, 5832, 20412, 739008, 12791896, 52394312, 52394312, 12791896, 739008, 20412, 69984, 4533744

Offset: 1

R. H. Hardin, Feb 20 2016

Table starts
......0.........4...........24.............108................432
......4........48..........384............2736..............18336
.....24.......384.........5888...........80112............1031344
....108......2736........80112.........2097552...........52394312
....432.....18336......1031344........52394312.........2563440512
...1620....118032.....12791896......1265974992.......121792778352
...5832....739008....154606864.....29881402560......5665397992608
..20412...4533744...1833130768....693021071760....259306140235672
..69984..27384288..21416076480..15854541802056..11718368891185840
.236196.163381968.247279304248.358760880894864.524154208020159720

			Some solutions for n=3 k=4
..3..3..3..2. .2..3..1..3. .1..1..2..0. .2..2..2..3. .3..1..2..3
..1..1..3..1. .2..3..3..2. .0..0..0..0. .0..0..2..0. .1..3..3..3
..3..1..1..1. .3..3..2..2. .1..1..0..2. .2..2..0..0. .3..1..1..3

R. H. Hardin, Table of n, a(n) for n = 1..219

Column 1 is A120908.

Empirical for column k:
k=1: a(n) = 6*a(n-1) -9*a(n-2)
k=2: a(n) = 10*a(n-1) -21*a(n-2) -20*a(n-3) -4*a(n-4) for n>5
k=3: [order 8] for n>9
k=4: [order 16] for n>17
k=5: [order 40] for n>41

A299989 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2k+1) = A152842(2n,2(n-k)) and T(n,2k) = A152842(2n,2(n-k)+1) for n >= k > 0.

Original entry on oeis.org

0, 1, 0, 3, 4, 1, 0, 9, 24, 22, 8, 1, 0, 27, 108, 171, 136, 57, 12, 1, 0, 81, 432, 972, 1200, 886, 400, 108, 16, 1, 0, 243, 1620, 4725, 7920, 8430, 5944, 2810, 880, 175, 20, 1, 0, 729, 5832, 20898, 44280, 61695, 59472, 40636, 19824, 6855, 1640, 258, 24, 1

Offset: 0

Franck Maminirina Ramaharo, Feb 26 2018

T(n,k) is the number of state diagrams having k components of n connected summed trefoil knots.

Row sums gives A001018.

			The triangle T(n, k) begins:
n\k 0     1      2      3       4       5       6      7        8       9
0:  0     1
1:  0     3      4      1
2:  0     9     24     22       8       1
3:  0    27    108    171     136      57      12       1
4:  0    81    432    972    1200     886     400     108      16       1

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.

Michael De Vlieger, Table of n, a(n) for n = 0..10301 (rows 0 <= n <= 100, flattened.)
Ryo Hanaki, Pseudo diagrams of knots, links and spatial graphs, Osaka Journal of Mathematics, Vol. 47 (2010), 863-883.
Louis H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Carolina Medina, Jorge Ramírez-Alfonsín and Gelasio Salazar, On the number of unknot diagrams, arXiv:1710.06470 [math.CO], 2017.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.

Row 2: row 5 of A158454.
Row 3: row 2 of A220665.
Row 4: row 5 of A219234.

row[n_] := CoefficientList[x*(x^2 + 4*x + 3)^n, x]; Array[row, 7, 0] // Flatten (* Jean-François Alcover, Mar 16 2018 *)

g(x, y) := taylor(x/(1 - y*(x^2 + 4*x + 3)), y, 0, 10)$
a : makelist(ratcoef(g(x, y), y, n), n, 0, 10)$
T : []$
for i:1 thru 11 do
  T : append(T, makelist(ratcoef(a[i], x, n), n, 0, 2*i - 1))$
T;

T(n, k) = polcoeff(x*(x^2 + 4*x + 3)^n, k);
tabf(nn) = for (n=0, nn, for (k=0, 2*n+1, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 03 2018

T(n,k) = coefficients of x*(x^2 + 4*x + 3)^n.
T(n,k) = T(n-1,k-2) + 4*T(n-1,k-1) + 3*T(n-1,k), with T(n,0) = 0, T(n,1) = 3^n and  T(n,2) =  4*n*3^(n-1).
T(n,n+k+1) = A152842(2*n,n+k) and T(n,n-k) = A152842(2*n,n+k+1), for n >= k >= 0.
T(n,1) =  A000244(n).
T(n,2) =  A120908(n).
T(n,n+1) = A069835(n).
T(n,2*n-1) = A139272(n).
T(n,2*n) = A008586(n).
T(n,2*n-2) = A140138(4*n) = A185872(2n,2) for n >= 1.
G.f.: x/(1 - y*(x^2 + 4*x + 3)).

Typo in row 6 corrected by Jean-François Alcover, Mar 16 2018

A203230 (n-1)-st elementary symmetric function of the first n terms of A010684.

Original entry on oeis.org

1, 4, 7, 24, 33, 108, 135, 432, 513, 1620, 1863, 5832, 6561, 20412, 22599, 69984, 76545, 236196, 255879, 787320, 846369, 2598156, 2775303, 8503056, 9034497, 27634932, 29229255, 89282088, 94065057, 286978140, 301327047, 918330048

Offset: 1

Clark Kimberling, Dec 30 2011

nonn

Cf. A010684, A203231, A120908 (bisection?).

r = {1, 3, 1, 3, 1, 3};
s = Flatten[{r, r, r, r, r, r, r, r, r}];
t[n_] := Part[s, Range[n]]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 32}]     (* A203230 *)

Conjecture: a(n)=6*a(n-2)-9*a(n-4) with G.f. x*(1+4*x+x^2) / (-1+3*x^2)^2 . - R. J. Mathar, Oct 15 2013

A203231 (n-1)-st elementary symmetric function of the first n terms of the periodic sequence (3,1,3,1,3,1,3,1,...).

Original entry on oeis.org

1, 4, 15, 24, 81, 108, 351, 432, 1377, 1620, 5103, 5832, 18225, 20412, 63423, 69984, 216513, 236196, 728271, 787320, 2421009, 2598156, 7971615, 8503056, 26040609, 27634932, 84499119, 89282088, 272629233, 286978140, 875283327, 918330048

Offset: 1

Clark Kimberling, Dec 30 2011

nonn

Cf. A010684, A203230, A120908 (bisection).

r = {3, 1, 3, 1, 3, 1};
s = Flatten[{r, r, r, r, r, r, r, r, r}];
t[n_] := Part[s, Range[n]]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 32}]     (* A203231 *)

Conjecture: a(n) = 6*a(n-2)-9*a(n-4) with G.f. x*(1+4*x+9*x^2) / (-1+3*x^2)^2 . - R. J. Mathar, Oct 15 2013

A120907 Triangle read by rows: T(n,k) is the number of ternary words of length n on {0,1,2} having sum of the lengths of the drops equal to k (n>=0, k>=0). The drops of a ternary word on {0,1,2} are the subwords 10,20 and 21, their lengths being the differences 1, 2 and 1, respectively.

Original entry on oeis.org

1, 3, 6, 2, 1, 10, 10, 7, 15, 30, 31, 4, 1, 21, 70, 105, 36, 11, 28, 140, 294, 184, 76, 6, 1, 36, 252, 714, 696, 396, 78, 15, 45, 420, 1554, 2160, 1666, 566, 141, 8, 1, 55, 660, 3102, 5808, 5918, 2990, 995, 136, 19, 66, 990, 5775, 13992, 18348, 12746, 5615, 1280, 226

Offset: 0

Emeric Deutsch, Jul 15 2006

Row n has 2*floor(n/2)+1 terms (i.e. each of the rows 2n and 2n+1 has 2n+1 terms). Row sums are the powers of 3 (A000244). T(n,0)=A000217(n+1) (the triangular numbers). Sum(k*T(n,k),k>=0)=4(n-1)3^(n-2)=A120908(n)=4*A027471(n).

			T(4,3)=4 because we have 1020,2010,2021 and 2120.
Triangle starts:
1;
3;
6,2,1;
10,10,7;
15,30,31,4,1;
21,70,105,36,11;

Cf. A000244, A000217, A120908, A027471, A120906.

G:=1/(1-z+t*z)/(1-2*z+z^2-t*z-t*z^2): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..2*floor(n/2)) od; # yields sequence in triangular form

G.f.=G(t,z)=1/[(1-z+tz)(1-2z+z^2-tz-tz^2)].

cp's OEIS Frontend

A269276 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally or antidiagonally adjacent neighbor totalling three exactly once.

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A269214 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally, diagonally or antidiagonally adjacent neighbor totalling three exactly once.

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A269097 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally or vertically adjacent neighbor totalling three exactly once.

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A269152 T(n,k) = Number of n X k 0..3 arrays with some element plus some horizontally, antidiagonally or vertically adjacent neighbor totalling three exactly once.

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A269186 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally, diagonally, antidiagonally or vertically adjacent neighbor totalling three exactly once.

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A299989 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) for n >= k > 0.

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A203230 (n-1)-st elementary symmetric function of the first n terms of A010684.

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A203231 (n-1)-st elementary symmetric function of the first n terms of the periodic sequence (3,1,3,1,3,1,3,1,...).

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A120907 Triangle read by rows: T(n,k) is the number of ternary words of length n on {0,1,2} having sum of the lengths of the drops equal to k (n>=0, k>=0). The drops of a ternary word on {0,1,2} are the subwords 10,20 and 21, their lengths being the differences 1, 2 and 1, respectively.

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A299989 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2k+1) = A152842(2n,2(n-k)) and T(n,2k) = A152842(2n,2(n-k)+1) for n >= k > 0.