A068293 -id:A068293 - cp's OEIS frontend

A151724 First differences of A151723.

0, 1, 6, 6, 18, 6, 18, 30, 42, 6, 18, 30, 54, 54, 42, 78, 90, 6, 18, 30, 54, 54, 54, 102, 150, 102, 42, 78, 138, 162, 114, 186, 186, 6, 18, 30, 54, 54, 54, 102, 150, 102, 54, 102, 174, 222, 198, 246, 342, 198, 42, 78, 138, 162, 162, 258, 402, 354, 162, 186

Offset: 0

David Applegate and N. J. A. Sloane, Jun 13 2009

nonn

			When written as a triangle:
0,
1,
6,
6,18,
6,18,30,42,
6,18,30,54,54,42,78,90,
6,18,30,54,54,54,102,150,102,42,78,138,162,114,186,186,
...
Right border gives 0 together with A068293. - _Omar E. Pol_, Mar 19 2015

S. M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 (see Example 6, page 224).

N. J. A. Sloane, Table of n, a(n) for n = 0..4095 [First 1024 terms from David Applegate and N. J. A. Sloane]
David Applegate, The movie version
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
David Applegate and N. J. A. Sloane, Table of n, A151724(n), A151723(n) for n = 0..1025
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.

Cf. A151723, A170898 (after dividing by 6), A170899, A169759.

A212209 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the square diagonal grid graph DG_(k,k).

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 24, 5, 0, 0, 0, 72, 120, 6, 0, 0, 0, 168, 6720, 360, 7, 0, 0, 0, 360, 935040, 126360, 840, 8, 0, 0, 0, 744, 325061760, 265035240, 1128960, 1680, 9, 0, 0, 0, 1512, 283192323840, 3322711053720, 17160407040, 6510000, 3024, 10

Offset: 1

Alois P. Heinz, May 04 2012

The square diagonal grid graph DG_(n,n) has n^2 = A000290(n) vertices and 2*(n-1)*(2*n-1) = A002943(n-1) edges; see A212208 for example. The chromatic polynomial of DG_(n,n) has n^2+1 = A002522(n) coefficients.

This graph is also called the king graph. - Andrew Howroyd, Jun 25 2017

			Square array A(n,k) begins:
  1,   0,       0,           0,                0, ...
  2,   0,       0,           0,                0, ...
  3,   0,       0,           0,                0, ...
  4,  24,      72,         168,              360, ...
  5, 120,    6720,      935040,        325061760, ...
  6, 360,  126360,   265035240,    3322711053720, ...
  7, 840, 1128960, 17160407040, 2949948395735040, ...

Andrew Howroyd, Table of n, a(n) for n = 1..153
Eric Weisstein's World of Mathematics, King Graph
Wikipedia, Chromatic Polynomial

Columns 1-5 give: A000027, A052762 = 24*A000332, 24*A068250, 24*A068251, 24*A068252.
Rows n=1-16 give: A000007, A000038, 3*A000007, 4*A068293, 5*A068294, 6*A068295, 7*A068296, 8*A068297, 9*A068298, 10*A068299, 11*A068300, 12*A068301, 13*A068302, 14*A068303, 15*A068304, 16*A068305.
Cf. A000290, A002943, A212208, A208021.

A091344 a(n) = 23^n - 32^n + 1.

Original entry on oeis.org

0, 1, 7, 31, 115, 391, 1267, 3991, 12355, 37831, 115027, 348151, 1050595, 3164071, 9516787, 28599511, 85896835, 257887111, 774054547, 2322950071, 6970423075, 20914414951, 62749536307, 188261191831, 564808741315, 1694476555591

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jan 01 2004

Starting with offset 1 = binomial transform of A068293: (1, 6, 18, 42, 90, ...) and double binomial transform of (1, 5, 7, 5, 7, 5, ...). - Gary W. Adamson, Jan 13 2009

Number of pairs (A,B) where A and B are nonempty subsets of {1,2,...,n} and one of these subsets is a subset of the other. - For the case that one of these subsets is a proper subset of the other see a(n+1) in A260217. - If empty subsets are included, see A027649 (all subsets) and A056182 (proper subsets). - Manfred Boergens, Aug 02 2023

Christian Ballot and Florian Luca, Prime factors of a^f(n)-1 with an irreducible polynomial f(x),New York J. Math. 12 (2006), 39-45 (electronic).
Christian Ballot and Florian Luca, Common prime factors of a^n-b and c^n-d, Unif. Distrib. Theory 2 (2007), no. 2, 19-34 (electronic).
John Elias, Illustration of initial terms: Sixfold Sierpinski Stars
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A027649, A056182, A068293, A260217.

a:=n->sum((3^(n-j-1)-2^(n-2-j))*12, j=0..n): seq(a(n), n=-1..24); # Zerinvary Lajos, Feb 11 2007
with (combinat):a:=n->stirling2(n,3)+stirling2(n+1,3): seq(a(n), n=1..26); # Zerinvary Lajos, Oct 07 2007

Table[Sum[i!i^2 StirlingS2[n, i](-1)^(n - i), {i, 1, n}], {n, 0, 30}]
Table[2*3^n-3*2^n+1,{n,0,30}] (* or *) LinearRecurrence[{6,-11,6},{0,1,7},30] (* Harvey P. Dale, Dec 31 2013 *)

a(n) = Sum_{i=1..n} i!*i^2*Stirling2(n,i)*(-1)^(n-i).
From Christian Ballot via R. K. Guy, Jan 13 2009: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3);
G.f.: x*(1+x)/((1-x)*(2-x)*(3-x)). (End)
a(n) = 5*a(n-1) - 6*a(n-2) + 2, a(0)=0, a(1)=1. - Vincenzo Librandi, Nov 25 2010
E.g.f.: exp(x)*(1 - 3*exp(x) + 2*exp(2*x)). - Stefano Spezia, May 18 2024

Edited by N. J. A. Sloane, Jan 13 2009 at the suggestion of R. K. Guy; the concise definition was provided by Vladeta Jovovic, Jan 01 2004

A175164 a(n) = 16*(2^n - 1).

Original entry on oeis.org

0, 16, 48, 112, 240, 496, 1008, 2032, 4080, 8176, 16368, 32752, 65520, 131056, 262128, 524272, 1048560, 2097136, 4194288, 8388592, 16777200, 33554416, 67108848, 134217712, 268435440

Offset: 0

Reinhard Zumkeller, Feb 28 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Sequences of the form m*(2^n - 1): A000225 (m=1), A000918 (m=2), A068156 (m=3), A028399 (m=4), A068293 (m=6), A159741 (m=8), this sequence (m=16), A175165 (m=32), A175166 (m=64).

Cf. A140504, A173787.

I:=[0,16]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021

16*(2^Range[0,40] - 1) (* G. C. Greubel, Jul 08 2021 *)

def A175164(n): return (1<Chai Wah Wu, Jun 27 2023

[16*(2^n -1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

a(n) = 2^(n+4) - 16.
a(n) = A173787(n+4, 4).
a(2*n) = A140504(n+2)*A028399(n).
a(n) = 3*a(n-1) - 2*a(n-2), a(0)=0, a(1)=16. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 16*x/((1-x)*(1-2*x)).
E.g.f.: 16*(exp(2*x) - exp(x)). (End)

A175166 a(n) = 64*(2^n - 1).

Original entry on oeis.org

0, 64, 192, 448, 960, 1984, 4032, 8128, 16320, 32704, 65472, 131008, 262080, 524224, 1048512, 2097088, 4194240, 8388544, 16777152, 33554368, 67108800, 134217664, 268435392, 536870848, 1073741760

Offset: 0

Reinhard Zumkeller, Feb 28 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Sequences of the form m*(2^n - 1): A000225 (m=1), A000918 (m=2), A068156 (m=3), A028399 (m=4), A068293 (m=6), A159741 (m=8), A175164 (m=16), A175165 (m=32), this sequence (m=64).

Cf. A173787, A175161.

I:=[0,64]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021

LinearRecurrence[{3,-2},{0,64},30] (* Harvey P. Dale, Apr 08 2015 *)

def A175166(n): return (1<Chai Wah Wu, Jun 27 2023

[64*(2^n -1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

a(n) = 2^(n+6) - 64.
a(n) = A173787(n+6, 6).
a(2*n) = A175161(n)*A159741(n) for n > 0.
a(n) = 3*a(n-1) - 2*a(n-2), a(0)=0, a(1)=64. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 64*x/((1-x)*(1-2*x)).
E.g.f.: 64*(exp(2*x) - exp(x)). (End)

A175165 a(n) = 32*(2^n - 1).

Original entry on oeis.org

0, 32, 96, 224, 480, 992, 2016, 4064, 8160, 16352, 32736, 65504, 131040, 262112, 524256, 1048544, 2097120, 4194272, 8388576, 16777184, 33554400, 67108832, 134217696, 268435424, 536870880

Offset: 0

Reinhard Zumkeller, Feb 28 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Sequences of the form m*(2^n - 1): A000225 (m=1), A000918 (m=2), A068156 (m=3), A028399 (m=4), A068293 (m=6), A159741 (m=8), A175164 (m=16), this sequence (m=32), A175166 (m=64).

Cf. A173787.

I:=[0,32]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021

32(2^Range[0,30] -1) (* or *) LinearRecurrence[{3,-2},{0,32},30] (* Harvey P. Dale, Mar 23 2015 *)

def A175165(n): return (1<Chai Wah Wu, Jun 27 2023

[32*(2^n -1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

a(n) = 2^(n+5) - 32.
a(n) = A173787(n+5, 5).
a(n) = 3*a(n-1) - 2*a(n-2); a(0)=0, a(1)=32. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 32*x/((1-x)*(1-2*x)).
E.g.f.: 32*(exp(2*x) - exp(x)). (End)

A212085 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the complete bipartite graph K_(k,k).

Original entry on oeis.org

0, 0, 2, 0, 2, 6, 0, 2, 18, 12, 0, 2, 42, 84, 20, 0, 2, 90, 420, 260, 30, 0, 2, 186, 1812, 2420, 630, 42, 0, 2, 378, 7332, 18500, 9750, 1302, 56, 0, 2, 762, 28884, 127220, 121590, 30702, 2408, 72, 0, 2, 1530, 112740, 825860, 1324470, 583422, 81032, 4104, 90

Offset: 1

Alois P. Heinz, Apr 30 2012

The complete bipartite graph K_(n,n) has 2*n vertices and n^2 = A000290(n) edges. The chromatic polynomial of K_(n,n) has 2*n+1 coefficients.

A(n,k) is the number of pairs of strings of length k over an alphabet of size n such that the strings do not share any letter. - Lin Zhangruiyu, Aug 19 2022

			A(3,1) = 6 because there are 6 3-colorings of the complete bipartite graph K_(1,1): 1-2, 1-3, 2-1, 2-3, 3-1, 3-2.
Square array A(n,k) begins:
   0,   0,    0,      0,       0,        0,         0, ...
   2,   2,    2,      2,       2,        2,         2, ...
   6,  18,   42,     90,     186,      378,       762, ...
  12,  84,  420,   1812,    7332,    28884,    112740, ...
  20, 260, 2420,  18500,  127220,   825860,   5191220, ...
  30, 630, 9750, 121590, 1324470, 13284630, 126657750, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Wikipedia, Chromatic Polynomial

Rows n=1-3 give: A000004, A007395, A068293(k+1).
Columns k=1-2 give: A002378(n-1), A091940.
Cf. A008277, A212084, A266695, A282245.

A:= (n, k)-> add(Stirling2(k, j) *mul(n-i, i=0..j-1) *(n-j)^k, j=1..k):
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);

a[n_, k_] := Sum[(-1)^j*(n-j)^k*Pochhammer[-n, j]*StirlingS2[k, j], {j, 1, k}]; Table[a[n-k, k], {n, 1, 11}, {k, n-1, 1, -1}] // Flatten (* Jean-François Alcover, Dec 11 2013 *)

A(n,k) = Sum_{j=1..k} (n-j)^k * S2(k,j) * Product_{i=0..j-1} (n-i).

A(n,n)/n = A282245(n).

A185739 Accumulation array of A185738, by antidiagonals.

Original entry on oeis.org

1, 3, 4, 6, 10, 11, 10, 18, 25, 26, 15, 28, 42, 56, 57, 21, 40, 62, 90, 119, 120, 28, 54, 85, 128, 186, 246, 247, 36, 70, 111, 170, 258, 378, 501, 502, 45, 88, 140, 216, 335, 516, 762, 1012, 1013, 55, 108, 172, 266, 417, 660, 1030, 1530, 2035, 2036, 66, 130, 207, 320, 504, 810, 1305, 2056, 3066, 4082, 4083, 78, 154, 245, 378, 596, 966, 1587, 2590, 4106, 6138, 8177, 8178, 91

Offset: 1

Clark Kimberling, Feb 02 2011

This arrays is a member of a chain; see A185738.

			Northwest corner:
1....3....6....10....15
4....10...18...28....40
11...25...42...62....85
26...56...90...128...170

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Rows 1 to 4: A000217, A028562, A140675, 2*A098847
Columns 1 to 3: A000295, A000247, A068293.
Cf. A185738, A185740.

(* See A185738 *)
f[n_, k_] := (k/2)*(4*(2^n - 1) + (k - 3)*n);
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 10}]]  (* Array A185739 *)
Table[f[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 11 2017 *)

T(n,k) = k*(4*(2^n-1)+(k-3)*n), k>=1, n>=1.

A383900 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of Product_{j=0..k} (1 + jx)/(1 - jx).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 2, 0, 1, 12, 18, 2, 0, 1, 20, 72, 42, 2, 0, 1, 30, 200, 312, 90, 2, 0, 1, 42, 450, 1400, 1152, 186, 2, 0, 1, 56, 882, 4650, 8000, 3912, 378, 2, 0, 1, 72, 1568, 12642, 38250, 40520, 12672, 762, 2, 0, 1, 90, 2592, 29792, 142002, 271770, 190400, 39912, 1530, 2, 0

Offset: 0

Seiichi Manyama, May 14 2025

			Square array begins:
  1, 1,   1,    1,     1,      1, ...
  0, 2,   6,   12,    20,     30, ...
  0, 2,  18,   72,   200,    450, ...
  0, 2,  42,  312,  1400,   4650, ...
  0, 2,  90, 1152,  8000,  38250, ...
  0, 2, 186, 3912, 40520, 271770, ...

Columns k=0..4 give A000007, A040000, A068293(n+1), A383910, A383911.
Main diagonal gives A350366.
A(n,n-1) gives A383767.
Cf. A383818, A383843.

a(n, k) = sum(j=0, k, abs(stirling(k+1, j+1, 1))*stirling(j+n, k, 2));

A(n,k) = Sum_{j=0..k} |Stirling1(k+1,j+1)| * Stirling2(j+n,k).

A178923 Rectangular array T(m,k)= StirlingS2(k-1,m-1)*m! (The Coupon Collectors Problem).

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 6, 0, 0, 0, 2, 18, 0, 0, 0, 0, 2, 42, 24, 0, 0, 0, 0, 2, 90, 144, 0, 0, 0, 0, 0, 2, 186, 600, 120, 0, 0, 0, 0, 0, 2, 378, 2160, 1200, 0, 0, 0, 0, 0

Offset: 1

Geoffrey Critzer, Dec 29 2010

T(m,k) is the number of functions f:{1,2,...}->{1,2,...,m} such that the image of f[{1,2,...,k}] is {1,2,...,m} but the image of f[{1,2,...,k-1}] is not.

T(m,k)/m^k is the probability that a collector of m different objects will require exactly k trials (uniform random selection with replacement) to complete the collection.

			   1   0   0   0   0     0     0      0       0 ...
   0   2   2   2   2     2     2      2       2 ...
   0   0   6  18  42    90   186    378     762 ...
   0   0   0  24 144   600  2160   7224   23184 ...
   0   0   0   0 120  1200  7800  42000  204120 ...
   0   0   0   0   0   720 10800 100800  756000 ...
   0   0   0   0   0     0  5040 105840 1340640 ...
   0   0   0   0   0     0     0  40320 1128960 ...
   0   0   0   0   0     0     0      0  362880 ...

Cf. A068293 (row m=3), A000142 (diagonal), A001804 (subdiagonal).

A178923 := proc(m,k)
    combinat[stirling2](k-1,m-1)*m! ;
end proc:
seq(seq(A178923(m,d-m),m=1..d-1),d=2..15) ; # R. J. Mathar, Jan 19 2024

Table[Table[StirlingS2[k - 1, m - 1] m!, {k, 1, 10}], {m, 1, 10}] // Grid

O.g.f. for row m: m!*x^m/Product_{i=1...m-1}1-i*x.

cp's OEIS Frontend

A151724 First differences of A151723.

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A212209 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the square diagonal grid graph DG_(k,k).

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A091344 a(n) = 2*3^n - 3*2^n + 1.

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A175164 a(n) = 16*(2^n - 1).

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A175166 a(n) = 64*(2^n - 1).

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A175165 a(n) = 32*(2^n - 1).

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Magma

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A212085 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the complete bipartite graph K_(k,k).

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Maple

Mathematica

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A185739 Accumulation array of A185738, by antidiagonals.

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A091344 a(n) = 23^n - 32^n + 1.

A383900 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of Product_{j=0..k} (1 + jx)/(1 - jx).