A120926 -id:A120926 - cp's OEIS frontend

A291000 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3.

1, 3, 9, 26, 74, 210, 596, 1692, 4804, 13640, 38728, 109960, 312208, 886448, 2516880, 7146144, 20289952, 57608992, 163568448, 464417728, 1318615104, 3743926400, 10630080640, 30181847168, 85694918912, 243312448256, 690833811712, 1961475291648, 5569190816256

Offset: 0

Clark Kimberling, Aug 22 2017

Suppose s = (c(0), c(1), c(2),...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x).  Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
In the following guide to p-INVERT sequences using s = (1,1,1,1,1,...) = A000012, in some cases t(1,1,1,1,1,...) is a shifted version of the cited sequence:
p(S)                             t(1,1,1,1,1,...)
1 - S                                A000079
1 - S^2                              A000079
1 - S^3                              A024495
1 - S^4                              A000749
1 - S^5                              A139761
1 - S^6                              A290993
1 - S^7                              A290994
1 - S^8                              A290995
1 - S - S^2                          A001906
1 - S - S^3                          A116703
1 - S - S^4                          A290996
1 - S^3 - S^6                        A290997
1 - S^2 - S^3                        A095263
1 - S^3 - S^4                        A290998
1 - 2 S^2                            A052542
1 - 3 S^2                            A002605
1 - 4 S^2                            A015518
1 - 5 S^2                            A163305
1 - 6 S^2                            A290999
1 - 7 S^2                            A291008
1 - 8 S^2                            A291001
(1 - S)^2                            A045623
(1 - S)^3                            A058396
(1 - S)^4                            A062109
(1 - S)^5                            A169792
(1 - S)^6                            A169793
(1 - S^2)^2                          A024007
1 - 2 S - 2 S^2                      A052530
1 - 3 S - 2 S^2                      A060801
(1 - S)(1 - 2 S)                     A053581
(1 - 2 S)(1 - 3 S)                   A291002
(1 - S)(1 - 2 S)(1 - 3 S)(1 - 4 S)   A291003
(1 - 2 S)^2                          A120926
(1 - 3 S)^2                          A291004
1 + S - S^2                          A000045  (Fibonacci numbers starting with -1)
1 - S - S^2 - S^3                    A291000
1 - S - S^2 - S^3 - S^4              A291006
1 - S - S^2 - S^3 - S^4 - S^5        A291007
1 - S^2 - S^4                        A290990
(1 - S)(1 - 3 S)                     A291009
(1 - S)(1 - 2 S)(1 - 3 S)            A291010
(1 - S)^2 (1 - 2 S)                  A291011
(1 - S^2)(1 - 2 S)                   A291012
(1 - S^2)^3                          A291013
(1 - S^3)^2                          A291014
1 - S - S^2 + S^3                    A045891
1 - 2 S - S^2 + S^3                  A291015
1 - 3 S + S^2                        A136775
1 - 4 S + S^2                        A291016
1 - 5 S + S^2                        A291017
1 - 6 S + S^2                        A291018
1 - S - S^2 - S^3 + S^4              A291019
1 - S - S^2 - S^3 - S^4 + S^5        A291020
1 - S - S^2 - S^3 + S^4 + S^5        A291021
1 - S - 2 S^2 + 2 S^3                A175658
1 - 3 S^2 + 2 S^3                    A291023
(1 - 2 S^2)^2                        A291024
(1 - S^3)^3                          A291143
(1 - S - S^2)^2                      A209917

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4, -4, 2)

Cf. A000012, A289780.

z = 60; s = x/(1 - x); p = 1 - s - s^2 - s^3;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291000 *)

G.f.: (-1 + x - x^2)/(-1 + 4 x - 4 x^2 + 2 x^3).

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

A269201 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally, diagonally or antidiagonally adjacent neighbor totalling three no more than once.

Original entry on oeis.org

4, 16, 16, 60, 180, 64, 216, 1284, 1740, 256, 756, 9612, 25572, 15540, 1024, 2592, 68052, 400428, 471492, 132300, 4096, 8748, 472044, 5877228, 15289548, 8314020, 1090740, 16384, 29160, 3212820, 84310620, 463790340, 555862380, 142233732

Offset: 1

R. H. Hardin, Feb 20 2016

Table starts
.......4.........16.............60...............216..................756
......16........180...........1284..............9612................68052
......64.......1740..........25572............400428..............5877228
.....256......15540.........471492..........15289548............463790340
....1024.....132300........8314020.........555862380..........34838403756
....4096....1090740......142233732.......19558138380........2532677348772
...16384....8787660.....2380537188......672230393004......179867149105740
...65536...69580980....39186271044....22702294138188....12551707872624132
..262144..543538380...636703584804...756261535626732...864008559706781292
.1048576.4200069300.10237337586180.24917784636315276.58827234014669683044

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

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

Column 1 is A000302.
Column 2 is A269103.
Row 1 is A120926(n+1).

Empirical for column k:
k=1: a(n) = 4*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

A269289 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally or antidiagonally adjacent neighbor totalling three no more than once.

Original entry on oeis.org

4, 16, 16, 60, 216, 64, 216, 2124, 2592, 256, 756, 19188, 62748, 29160, 1024, 2592, 164556, 1363572, 1698732, 314928, 4096, 8748, 1363572, 27788292, 87559668, 43674876, 3306744, 16384, 29160, 11026764, 544118148, 4204943820, 5306911092

Offset: 1

R. H. Hardin, Feb 21 2016

Table starts
......4.........16.............60...............216...................756
.....16........216...........2124.............19188................164556
.....64.......2592..........62748...........1363572..............27788292
....256......29160........1698732..........87559668............4204943820
...1024.....314928.......43674876........5306911092..........598478857956
...4096....3306744.....1085203980......309846524148........81907569617580
..16384...34012224....26317946844....17623065834612.....10908770041709316
..65536..344373768...626778812268...983118947312628...1424067311317705740
.262144.3443737680.14718495557052.54032675767734132.183070424003703987492

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

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

Column 1 is A000302.
Column 2 is A159739(n+1).
Row 1 is A120926(n+1).

Empirical for column k:
k=1: a(n) = 4*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

A269109 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally or vertically adjacent neighbor totalling three no more than once.

Original entry on oeis.org

4, 16, 16, 60, 180, 60, 216, 1740, 1740, 216, 756, 15540, 40908, 15540, 756, 2592, 132300, 872460, 872460, 132300, 2592, 8748, 1090740, 17593092, 43964700, 17593092, 1090740, 8748, 29160, 8787660, 342055548, 2085484068, 2085484068

Offset: 1

R. H. Hardin, Feb 19 2016

Table starts
......4.........16.............60................216....................756
.....16........180...........1740..............15540.................132300
.....60.......1740..........40908.............872460...............17593092
....216......15540.........872460...........43964700.............2085484068
....756.....132300.......17593092.........2085484068...........232068730044
...2592....1090740......342055548........95166487524.........24808345933548
...8748....8787660.....6482020140......4227147007836.......2579398703502996
..29160...69580980...120520189980....184069947098892.....262780733311913580
..96228..543538380..2208175854948...7894012975085748...26357371124964908460
.314928.4200069300.39988864047276.334480929126425748.2611360040338484328156

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

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

Column 1 is A120926(n+1).

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

A269143 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally, antidiagonally or vertically adjacent neighbor totalling three no more than once.

Original entry on oeis.org

4, 16, 16, 60, 148, 60, 216, 1164, 1164, 216, 756, 8532, 18556, 8532, 756, 2592, 59916, 275796, 275796, 59916, 2592, 8748, 408596, 3924212, 8317996, 3924212, 408596, 8748, 29160, 2727564, 54199284, 240647068, 240647068, 54199284, 2727564, 29160

Offset: 1

R. H. Hardin, Feb 20 2016

Table starts
......4........16............60..............216.................756
.....16.......148..........1164.............8532...............59916
.....60......1164.........18556...........275796.............3924212
....216......8532........275796..........8317996...........240647068
....756.....59916.......3924212........240647068.........14197016484
...2592....408596......54199284.......6766301156........815458664276
...8748...2727564.....732561916.....186315931804......45920055321732
..29160..17914580....9740150372....5049212790572....2546667557472940
..96228.116170764..127846717716..135126561336764..139535357045338964
.314928.745617300.1660741102212.3579710280903028.7570570235882777884

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

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

Column 1 is A120926(n+1).

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

A269194 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally, diagonally, antidiagonally or vertically adjacent neighbor totalling three no more than once.

Original entry on oeis.org

4, 16, 16, 60, 108, 60, 216, 708, 708, 216, 756, 4476, 9284, 4476, 756, 2592, 27684, 115452, 115452, 27684, 2592, 8748, 168252, 1399612, 2817548, 1399612, 168252, 8748, 29160, 1008804, 16629436, 67134380, 67134380, 16629436, 1008804, 29160

Offset: 1

R. H. Hardin, Feb 20 2016

Table starts
......4........16...........60.............216................756
.....16.......108..........708............4476..............27684
.....60.......708.........9284..........115452............1399612
....216......4476.......115452.........2817548...........67134380
....756.....27684......1399612........67134380.........3159954052
...2592....168252.....16629436......1567933676.......145995171708
...8748...1008804....194596516.....36068275268......6648192704604
..29160...5983164...2249848972....819789099172....299224791093196
..96228..35170980..25758552060..18452018423420..13339889117822420
.314928.205214268.292530733516.411983259384452.590022908205332548

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

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

Column 1 is A120926(n+1).

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

A120924 Triangle read by rows: T(n,k) is the number of ternary words of length n on {0,1,2}, having k isolated 0's (n >= 0, k >= 0).

Original entry on oeis.org

1, 2, 1, 5, 4, 13, 12, 2, 33, 36, 12, 83, 108, 48, 4, 209, 316, 172, 32, 527, 904, 588, 160, 8, 1329, 2548, 1932, 672, 80, 3351, 7104, 6140, 2592, 480, 16, 8449, 19628, 19020, 9440, 2320, 192, 21303, 53816, 57756, 32896, 10000, 1344, 32, 53713, 146596

Offset: 0

Emeric Deutsch, Jul 16 2006

Row n has 1+ceiling(n/2) terms.
Row sums are the powers of 3 (A000244).
T(n,0) = A120925(n).
Sum_{k=0..ceiling(n/2)} k*T(n,k) = A120926(n).

			T(2,0)=5 because we have 00,11,12,21 and 22; T(2,1)=4 because we have 01,02,10 and 20; T(3,2)=2 because we have 010 and 020.
Triangle starts:
   1;
   2,   1;
   5,   4;
  13,  12,  2;
  33,  36, 12;
  83, 108, 48, 4;

Cf. A000244, A105114, A120925, A120926.

G:=(1-(1-t)*z*(1-z))/(1-3*z+2*(1-t)*z^2*(1-z)): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 13 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 13 do seq(coeff(P[n],t,j),j=0..ceil(n/2)) od; # yields sequence in triangular form

G.f. = G(t,z) = (1-(1-t)z(1-z))/(1 - 3z + 2(1-t)z^2*(1-z)).

A123515 Triangle read by rows: T(n,k) is the number of involutions of {1,2,...,n} with exactly k fixed points and which contain the pattern 231 exactly once (n>=4, 2<=k<=n-2).

Original entry on oeis.org

1, 0, 2, 2, 0, 3, 0, 8, 0, 4, 5, 0, 18, 0, 5, 0, 26, 0, 32, 0, 6, 12, 0, 75, 0, 50, 0, 7, 0, 76, 0, 164, 0, 72, 0, 8, 28, 0, 264, 0, 305, 0, 98, 0, 9, 0, 208, 0, 680, 0, 510, 0, 128, 0, 10, 64, 0, 840, 0, 1460, 0, 791, 0, 162, 0, 11, 0, 544, 0, 2480, 0, 2772, 0, 1160, 0, 200, 0, 12

Offset: 4

Emeric Deutsch, Oct 13 2006

Also the number of involutions of {1,2,...,n} with exactly k fixed points and which contain the pattern 312 exactly once (n>=4, 2<=k<=n-2). Example: T(5,3)=2 because we have 15342 and 42315 (also the involution 52341 has 3 fixed points but it contains 3 times the pattern 312: 523, 524 and 534).

			T(5,3)=2 because we have 15342 and 42315 (also the involution 52341 has 3 fixed points but it contains 3 times the pattern 231: 231, 241 and 341).
Triangle starts:
   1;
   0,   2;
   2,   0,   3;
   0,   8,   0,    4;
   5,   0,  18,    0,    5;
   0,  26,   0,   32,    0,    6;
  12,   0,  75,    0,   50,    0,   7;
   0,  76,   0,  164,    0,   72,   0,    8;
  28,   0, 264,    0,  305,    0,  98,    0,   9;
   0, 208,   0,  680,    0,  510,   0,  128,   0,  10;
  64,   0, 840,    0, 1460,    0, 791,    0, 162,   0, 11;
   0, 544,   0, 2480,    0, 2772,   0, 1160,   0, 200,  0, 12;

G. C. Greubel, Rows n = 4..54 of the triangle, flattened
E. Deutsch, A. Robertson and D. Saracino, Refined restricted involutions, European Journal of Combinatorics 28 (2007), 481-498 (see pp. 492 and 498).

Cf. A045623, A120926, A123514, A112554.

T:=proc(n,k) if n>=4 and n+k mod 2 = 0 then (k-1)*2^((n-k-6)/2)*(binomial((n+k)/2-2,(n-k)/2-1)+2*binomial((n+k)/2-3, (n-k)/2-1)+binomial((n+k)/2-4,(n-k)/2-1)) else 0 fi end: for n from 4 to 16 do seq(T(n,k),k=2..n-2) od; # yields sequence in triangular form

T[n_, k_]:= ((1+(-1)^(n-k))/2)*2^((n-k-6)/2)*(k-1)* Sum[Binomial[2, j]*
  Binomial[(n+k-2*(j+2))/2, (n-k-2)/2], {j, 0, 2}];
Table[T[n, k], {n,4,16}, {k,2,n-2}]//Flatten (* G. C. Greubel, Jan 16 2022 *)

def A123515(n,k): return ((1+(-1)^(n+k))/2)*2^((n-k-6)/2)*(k-1)*sum( binomial(2, j)*binomial((n+k-2*j-2)/2, (n-k-2)/2) for j in (0..2) )
flatten([[A123515(n,k) for k in (2..n-2)] for n in (4..16)]) # G. C. Greubel, Jan 16 2022

T(n, k) = 2^((n-k-6)/2)*(k-1)*( binomial((n+k)/2-2, (n-k)/2-1) + 2*binomial((n+k)/2-3, (n-k)/2-1) + binomial((n+k)/2-4, (n-k)/2-1) ) for n>=4, n+k even; T(n,k) = 0 otherwise.
From G. C. Greubel, Jan 16 2022: (Start)
Sum_{k=2..n-4} T(n, k) = A045623(n).
Sum_{k=2..floor(n/2)} T(n-k+2, k) = (1/9)*[n=4] + (1+(-1)^n)*n*3^((n-8)/2). (End)

A209240 Triangular array read by rows. T(n,k) is the number of ternary length-n words in which the longest run of consecutive 0's is exactly k; n>=0, 0<=k<=n.

Original entry on oeis.org

1, 2, 1, 4, 4, 1, 8, 14, 4, 1, 16, 44, 16, 4, 1, 32, 132, 58, 16, 4, 1, 64, 384, 200, 60, 16, 4, 1, 128, 1096, 668, 214, 60, 16, 4, 1, 256, 3088, 2180, 740, 216, 60, 16, 4, 1, 512, 8624, 6992, 2504, 754, 216, 60, 16, 4, 1, 1024, 23936, 22128, 8332, 2576, 756, 216, 60, 16, 4, 1

Offset: 0

Geoffrey Critzer, Jan 13 2013

Row sums are 3^n.
Column k=0 is A000079.
Column k=1 is A094309.
Limit of reversed rows gives A120926.

			1;
2,   1;
4,   4,    1;
8,   14,   4,    1;
16,  44,   16,   4,   1;
32,  132,  58,   16,  4,   1;
64,  384,  200,  60,  16,  4,  1;
128, 1096, 668,  214, 60,  16, 4,  1;
256, 3088, 2180, 740, 216, 60, 16, 4,  1;

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A048004.

nn=10;f[list_]:=Select[list,#>0&];Map[f,Transpose[Table[CoefficientList[ Series[(1-x^k)/(1-3x+2x^(k+1))-(1-x^(k-1))/(1-3x+2x^k),{x,0,nn}],x],{k,1,nn+1}]]]//Grid

O.g.f. for column k: (1-x)^2*x^k/(1-3*x+2*x^(k+1))/(1-3*x+2*x^(k+2)).

cp's OEIS Frontend

A291000 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - S - S^2 - S^3.

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Mathematica

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A269201 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally, diagonally or antidiagonally adjacent neighbor totalling three no more than once.

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A269289 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally or antidiagonally adjacent neighbor totalling three no more than once.

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A269109 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally or vertically adjacent neighbor totalling three no more than once.

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A269143 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally, antidiagonally or vertically adjacent neighbor totalling three no more than once.

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A269194 T(n,k)=Number of nXk 0..3 arrays with some element plus some horizontally, diagonally, antidiagonally or vertically adjacent neighbor totalling three no more than once.

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A120924 Triangle read by rows: T(n,k) is the number of ternary words of length n on {0,1,2}, having k isolated 0's (n >= 0, k >= 0).

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Maple

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A123515 Triangle read by rows: T(n,k) is the number of involutions of {1,2,...,n} with exactly k fixed points and which contain the pattern 231 exactly once (n>=4, 2<=k<=n-2).

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Sage

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A209240 Triangular array read by rows. T(n,k) is the number of ternary length-n words in which the longest run of consecutive 0's is exactly k; n>=0, 0<=k<=n.