A170733 -id:A170733 - cp's OEIS frontend

A003945 Expansion of g.f. (1+x)/(1-2*x).

1, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888

Offset: 0

N. J. A. Sloane

Coordination sequence for infinite tree with valency 3.
Number of Hamiltonian cycles in K_3 X P_n.
Number of ternary words of length n avoiding aa, bb, cc.
For n > 0, row sums of A029635. - Paul Barry, Jan 30 2005
Binomial transform is {1, 4, 13, 40, 121, 364, ...}, see A003462. - Philippe Deléham, Jul 23 2005
Convolved with the Jacobsthal sequence A001045 = A001786: (1, 4, 12, 32, 80, ...). - Gary W. Adamson, May 23 2009
Equals (n+1)-th row sums of triangle A161175. - Gary W. Adamson, Jun 05 2009
a(n) written in base 2: a(0) = 1, a(n) for n >= 1: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-1) times 0 (see A003953(n)). - Jaroslav Krizek, Aug 17 2009
INVERTi transform of A003688. - Gary W. Adamson, Aug 05 2010
An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 42, 138, 162 and 168, lead to this sequence. For the corner squares these vectors lead to the companion sequence A083329. - Johannes W. Meijer, Aug 15 2010
A216022(a(n)) != 2 and A216059(a(n)) != 3. - Reinhard Zumkeller, Sep 01 2012
Number of length-n strings of 3 letters with no two adjacent letters identical. The general case (strings of r letters) is the sequence with g.f. (1+x)/(1-(r-1)*x). - Joerg Arndt, Oct 11 2012
Sums of pairs of rows of Pascal's triangle A007318, T(2n,k)+T(2n+1,k); Sum_{n>=1} A000290(n)/a(n) = 4. - John Molokach, Sep 26 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Yasemin Alp and E. Gokcen Kocer, Exponential Almost-Riordan Arrays, Results Math. (2024) Vol. 79, 173.
F. Faase, Counting Hamiltonian cycles in product graphs
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 151
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 304
Markus Kuba and Alois Panholzer, Enumeration formulas for pattern restricted Stirling permutations, Discrete Math. 312 (2012), no. 21, 3179--3194. MR2957938. - From _N. J. A. Sloane_, Sep 25 2012
C. Richard and U. Grimm, On the entropy and letter frequencies of ternary squarefree words, arXiv:math/0302302 [math.CO], 2003.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (2).
Index entries for sequences related to trees

Essentially same as A007283 (3*2^n) and A042950.
Cf. A001787, A001045, A161175.
Generating functions of the form (1+x)/(1-k*x) for k=1 to 12: A040000, A003945, A003946, A003947, A003948, A003949, A003950, A003951, A003952.
Generating functions of the form (1+x)/(1-k*x) for k=13 to 30: A170732, A170733, A170734, A170735, A170736, A170737, A170738, A170739, A170740, A170741, A170742, A170743, A170744, A170745, A170746, A170747, A170748.
Generating functions of the form (1+x)/(1-k*x) for k=31 to 50: A170749, A170750, A170751, A170752, A170753, A170754, A170755, A170756, A170757, A170758, A170759, A170760, A170761, A170762, A170763, A170764, A170765, A170766, A170767, A170768, A170769.
Cf. A003688.

k := 3; if n = 0 then 1 else k*(k-1)^(n-1); fi;

Join[{1}, 3*2^Range[0, 60]] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
Table[2^n+Floor[2^(n-1)], {n,0,30}] (* Martin Grymel, Oct 17 2012 *)
CoefficientList[Series[(1+x)/(1-2x),{x,0,40}],x] (* or *) LinearRecurrence[ {2},{1,3},40] (* Harvey P. Dale, May 04 2017 *)

a(n)=if(n,3<Charles R Greathouse IV, Jan 12 2012

a(0) = 1; for n > 0, a(n) = 3*2^(n-1).
a(n) = 2*a(n-1), n > 1; a(0)=1, a(1)=3.
More generally, the g.f. (1+x)/(1-k*x) produces the sequence [1, 1 + k, (1 + k)*k, (1 + k)*k^2, ..., (1+k)*k^(n-1), ...], with a(0) = 1, a(n) = (1+k)*k^(n-1) for n >= 1. Also a(n+1) = k*a(n) for n >= 1. - Zak Seidov and N. J. A. Sloane, Dec 05 2009
The g.f. (1+x)/(1-k*x) produces the sequence with closed form (in PARI notation) a(n)=(n>=0)*k^n+(n>=1)*k^(n-1). - Jaume Oliver Lafont, Dec 05 2009
Binomial transform of A000034. a(n) = (3*2^n - 0^n)/2. - Paul Barry, Apr 29 2003
a(n) = Sum_{k=0..n} (n+k)*binomial(n, k)/n. - Paul Barry, Jan 30 2005
a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 1. - Philippe Deléham, Jul 10 2005
Binomial transform of A000034. Hankel transform is {1,-3,0,0,0,...}. - Paul Barry, Aug 29 2006
a(0) = 1, a(n) = 2 + Sum_{k=0..n-1} a(k) for n >= 1. - Joerg Arndt, Aug 15 2012
a(n) = 2^n + floor(2^(n-1)). - Martin Grymel, Oct 17 2012
E.g.f.: (3*exp(2*x) - 1)/2. - Stefano Spezia, Jan 31 2023

Edited by N. J. A. Sloane, Dec 04 2009

A170734 Expansion of g.f.: (1+x)/(1-14*x).

Original entry on oeis.org

1, 15, 210, 2940, 41160, 576240, 8067360, 112943040, 1581202560, 22136835840, 309915701760, 4338819824640, 60743477544960, 850408685629440, 11905721598812160, 166680102383370240, 2333521433367183360, 32669300067140567040, 457370200939967938560, 6403182813159551139840

Offset: 0

N. J. A. Sloane, Dec 04 2009

For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,...,14} with no two adjacent letters identical. -Milan Janjic, Jan 31 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..800
M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, J. Int. Seq. 18 (2015) # 15.4.7.
Index entries for linear recurrences with constant coefficients, signature (14).

Cf. A003945, A003954, A097805, A170733.

k:=15;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

k:=15; [1] cat [k*(k-1)^(n-1): n in [1..25]]; // G. C. Greubel, Sep 24 2019

k:=15; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # G. C. Greubel, Sep 24 2019

Join[{1}, 15*14^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
CoefficientList[Series[(1+x)/(1-14x), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 10 2012 *)

vector(26, n, k=15; if(n==1, 1, k*(k-1)^(n-2))) \\ G. C. Greubel, Sep 24 2019

k=15; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Sep 24 2019

a(n) = Sum_{k=0..n} A097805(n,k)*(-1)^(n-k)*15^k. - Philippe Deléham, Dec 04 2009
a(0) = 1; for n>0, a(n) = 15*14^(n-1). - Vincenzo Librandi, Dec 05 2009
a(0) = 1, a(1) = 15, a(n) = 14*a(n-1). - Vincenzo Librandi, Dec 10 2012
E.g.f.: (15*exp(14*x) -1)/14. - G. C. Greubel, Sep 24 2019

A167922 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

Original entry on oeis.org

1, 14, 182, 2366, 30758, 399854, 5198102, 67575326, 878479238, 11420230094, 148462991222, 1930018885886, 25090245516518, 326173191714734, 4240251492291542, 55123269399790046, 716602502197270507

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

nonn

The initial terms coincide with those of A170733, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,-78).

Cf. A154638, A169452, A170733.

Cf. A167881, A167882, A167896 - A167900, A167908, A167914, A167916, A167919, A167923, A167924, A167926, A167927, A167929, A167931, A167933, A167935, A167937, A167938, A167940 - A167947, A167949 - A167962, A167978, A167980, A167988, A167989.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-13*x+90*x^16-78*x^17) )); // G. C. Greubel, Sep 13 2023

CoefficientList[Series[(1+t)*(1-t^16)/(1-13*t+90*t^16-78*t^17), {t, 0, 50}], t] (* G. C. Greubel, Jul 01 2016; Sep 13 2023 *)
coxG[{16, 78, -12, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 13 2023 *)

def A167922_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^16)/(1-13*x+90*x^16-78*x^17) ).list()
A167922_list(40) # G. C. Greubel, Sep 13 2023

G.f.: (t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/( 78*t^16 - 12*t^15 - 12*t^14 - 12*t^13 - 12*t^12 - 12*t^11 - 12*t^10 - 12*t^9 - 12*t^8 - 12*t^7 - 12*t^6 - 12*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).
From G. C. Greubel, Sep 13 2023: (Start)
G.f.: (1+t)*(1-t^16)/(1 - 13*t + 90*t^16 - 78*t^17).
a(n) = 12*Sum_{j=1..15} a(n-j) - 78*a(n-16). (End)

A158497 Triangle T(n,k) formed by the coordination sequences and the number of leaves for trees.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 12, 1, 4, 12, 36, 108, 1, 5, 20, 80, 320, 1280, 1, 6, 30, 150, 750, 3750, 18750, 1, 7, 42, 252, 1512, 9072, 54432, 326592, 1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 1, 9, 72, 576, 4608, 36864, 294912, 2359296, 18874368, 150994944, 1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, 430467210, 3874204890

Offset: 0

Thomas Wieder, Mar 20 2009

Consider the k-fold Cartesian products CP(n,k) of the vector A(n) = [1, 2, 3, ..., n].
An element of CP(n,k) is a n-tuple T_t of the form T_t = [i_1, i_2, i_3, ..., i_k] with t=1, .., n^k.
We count members T of CP(n,k) which satisfy some condition delta(T_t), so delta(.) is an indicator function which attains values of 1 or 0 depending on whether T_t is to be counted or not; the summation sum_{CP(n,k)} delta(T_t) over all elements T_t of CP produces the count.
For the triangle here we have delta(T_t) = 0 if for any two i_j, i_(j+1) in T_t one has i_j = i_(j+1): T(n,k) = Sum_{CP(n,k)} delta(T_t) = Sum_{CP(n,k)} delta(i_j = i_(j+1)).
The test on i_j > i_(j+1) generates A158498. One gets the Pascal triangle A007318 if the indicator function tests whether for any two i_j, i_(j+1) in T_t one has i_j >= i_(j+1).
Use of other indicator functions can also calculate the Bell numbers A000110, A000045 or A000108.

			Array, A(n, k) = n*(n-1)^(k-1) for n > 1, A(n, k) = 1 otherwise, begins as:
  1,  1,   1,    1,     1,      1,       1,        1,        1, ... A000012;
  1,  1,   1,    1,     1,      1,       1,        1,        1, ... A000012;
  1,  2,   2,    2,     2,      2,       2,        2,        2, ... A040000;
  1,  3,   6,   12,    24,     48,      96,      192,      384, ... A003945;
  1,  4,  12,   36,   108,    324,     972,     2916,     8748, ... A003946;
  1,  5,  20,   80,   320,   1280,    5120,    20480,    81920, ... A003947;
  1,  6,  30,  150,   750,   3750,   18750,    93750,   468750, ... A003948;
  1,  7,  42,  252,  1512,   9072,   54432,   326592,  1959552, ... A003949;
  1,  8,  56,  392,  2744,  19208,  134456,   941192,  6588344, ... A003950;
  1,  9,  72,  576,  4608,  36864,  294912,  2359296, 18874368, ... A003951;
  1, 10,  90,  810,  7290,  65610,  590490,  5314410, 47829690, ... A003952;
  1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, ............. A003953;
  1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, ............. A003954;
  1, 13, 156, 1872, 22464, 269568, 3234816, 38817792, ............. A170732;
  ... ;
The triangle begins as:
  1
  1, 1;
  1, 2,  2;
  1, 3,  6,  12;
  1, 4, 12,  36,  108;
  1, 5, 20,  80,  320,  1280;
  1, 6, 30, 150,  750,  3750,  18750;
  1, 7, 42, 252, 1512,  9072,  54432, 326592;
  1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344;
  ...;
T(3,3) = 12 counts the triples (1,2,1), (1,2,3), (1,3,1), (1,3,2), (2,1,2), (2,1,3), (2,3,1), (2,3,2), (3,1,2), (3,1,3), (3,2,1), (3,2,3) out of a total of 3^3 = 27 triples in the CP(3,3).

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Array rows n: A170733 (n=14), ..., A170769 (n=50).
Columns k: A000012(n) (k=0), A000027(n) (k=1), A002378(n-1) (k=2), A011379(n-1) (k=3), A179824(n) (k=4), A101362(n-1) (k=5), 2*A168351(n-1) (k=6), 2*A168526(n-1) (k=7), 2*A168635(n-1) (k=8), 2*A168675(n-1) (k=9), 2*A170783(n-1) (k=10), 2*A170793(n-1) (k=11).
Diagonals k: A055897 (k=n), A055541 (k=n-1), A373395 (k=n-2), A379612 (k=n-3).
Sums: (-1)^n*A065440(n) (signed row).
Cf. A000045, A000108, A000110, A007318, A079901, A158498.

A158497:= func< n,k | k le 1 select n^k else n*(n-1)^(k-1) >;
[A158497(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 18 2025

A158497[n_, k_]:= If[n<2 || k==0, 1, n*(n-1)^(k-1)];
Table[A158497[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 18 2025 *)

def A158497(n,k): return n^k if k<2 else n*(n-1)^(k-1)
print(flatten([[A158497(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 18 2025

T(n, k) = (n-1)^(k-1) + (n-1)^k = n*A079901(n-1,k-1), k > 0.

Sum_{k=0..n} T(n,k) = (n*(n-1)^n - 2)/(n-2), n > 2.

Edited by R. J. Mathar, Mar 31 2009

More terms added by G. C. Greubel, Mar 18 2025

A162783 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

Original entry on oeis.org

1, 14, 182, 2275, 28392, 353808, 4408950, 54938520, 684572616, 8530235532, 106292493216, 1324476080928, 16503864518232, 205649272719072, 2562528512535264, 31930831990629936, 397879682765894784

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

nonn

The initial terms coincide with those of A170733, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (12, 12, -78).

a:=[14, 182, 2275];; for n in [4..20] do a[n]:=12*a[n-1]+12*a[n-2] - 78*a[n-3]; od; Concatenation([1], a); # G. C. Greubel, Apr 26 2019

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^3)/(1-13*x+90*x^3-78*x^4) )); // G. C. Greubel, Apr 26 2019

CoefficientList[Series[(1+x)*(1-x^3)/(1-13*x+90*x^3-78*x^4), {x,0,20}],x] (* or *) coxG[{3, 78, -12}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 26 2019 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^3)/(1-13*x+90*x^3-78*x^4)) \\ G. C. Greubel, Apr 26 2019

((1+x)*(1-x^3)/(1-13*x+90*x^3-78*x^4)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019

G.f.: (t^3 + 2*t^2 + 2*t + 1)/(78*t^3 - 12*t^2 - 12*t + 1).

G.f.: (1+x)*(1-x^3)/(1 - 13*x + 90*x^3 - 78*x^4). - G. C. Greubel, Apr 26 2019

A166379 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

Original entry on oeis.org

1, 14, 182, 2366, 30758, 399854, 5198102, 67575326, 878479238, 11420230094, 148462991222, 1930018885795, 25090245514152, 326173191668688, 4240251491494200, 55123269386840928, 716602501995344328

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

The initial terms coincide with those of A170733, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (12, 12, 12, 12, 12, 12, 12, 12, 12, 12, -78).

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^11)/(1-13*x+90*x^11-78*x^12) )); // G. C. Greubel, Apr 26 2019

CoefficientList[Series[(1+x)*(1-x^11)/(1 -13*x +90*x^11 -78*x^12), {x, 0, 20}], x] (* G. C. Greubel, May 10 2016, modified Apr 26 2019 *)
coxG[{11,78,-12}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Nov 30 2016 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^11)/(1-13*x+90*x^11-78*x^12)) \\ G. C. Greubel, Apr 26 2019

((1+x)*(1-x^11)/(1-13*x+90*x^11-78*x^12)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019

G.f.: (t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(78*t^11 - 12*t^10 - 12*t^9 - 12*t^8 - 12*t^7 - 12*t^6 - 12*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).
G.f.: (1+x)*(1-x^11)/(1 -13*x +90*x^11 -78*x^12). - G. C. Greubel, Apr 26 2019
a(n) = -78*a(n-11) + 12*Sum_{k=1..10} a(n-k). - Wesley Ivan Hurt, May 06 2021

A163439 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

Original entry on oeis.org

1, 14, 182, 2366, 30758, 399763, 5195736, 67529280, 877681896, 11407280976, 148261073142, 1926957516120, 25044775341768, 325508355356184, 4230650423530440, 54986001777229068, 714656161291232160

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

The initial terms coincide with those of A170733, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..895
Index entries for linear recurrences with constant coefficients, signature (12,12,12,12,-78).

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-13*x+90*x^5-78*x^6) )); // G. C. Greubel, May 12 2019

CoefficientList[Series[(1+x)*(1-x^5)/(1-13*x+90*x^5-78*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{12, 12, 12, 12, -78}, {1, 14, 182, 2366, 30758, 399763}, 30] (* G. C. Greubel, Dec 23 2016 *)
coxG[{5, 78, -12}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 12 2019 *)

my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-13*x+90*x^5-78*x^6)) \\ G. C. Greubel, Dec 23 2016

((1+x)*(1-x^5)/(1-13*x+90*x^5-78*x^6)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(78*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).

a(n) = 12*a(n-1)+12*a(n-2)+12*a(n-3)+12*a(n-4)-78*a(n-5). - Wesley Ivan Hurt, May 10 2021

A163959 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

Original entry on oeis.org

1, 14, 182, 2366, 30758, 399854, 5198011, 67572960, 878433192, 11419432752, 148450042104, 1929816959616, 25087183842630, 326127713832648, 4239586491528024, 55113665158705608, 716465177275138200

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

nonn

The initial terms coincide with those of A170733, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..895
Index entries for linear recurrences with constant coefficients, signature (12,12,12,12,12,-78).

a:=[14, 182, 2366, 30758, 399854, 5198011];; for n in [7..30] do a[n]:=12*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -78*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 11 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-13*t+90*t^6-78*t^7) )); // G. C. Greubel, Aug 11 2019

seq(coeff(series((1+t)*(1-t^6)/(1-13*t+90*t^6-78*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 11 2019

CoefficientList[Series[(1+t)*(1-t^6)/(1-13*t+90*t^6-78*t^7), {t,0,30}], t] (* G. C. Greubel, Aug 13 2017 *)
coxG[{6, 78, -12}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 11 2019 *)

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-13*t+90*t^6-78*t^7)) \\ G. C. Greubel, Aug 13 2017

def A163959_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-13*t+90*t^6-78*t^7)).list()
A163959_list(30) # G. C. Greubel, Aug 11 2019

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(78*t^6 - 12*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).

a(n) = -78*a(n-6) + 12*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021

A164618 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

Original entry on oeis.org

1, 14, 182, 2366, 30758, 399854, 5198102, 67575235, 878476872, 11420184048, 148462193880, 1930005936768, 25090043590248, 326170130032656, 4240206014105334, 55122604391319192, 716592897791781192

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

nonn

The initial terms coincide with those of A170733, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..895
Index entries for linear recurrences with constant coefficients, signature (12,12,12,12,12,12,-78).

a:=[14,182,2366,30758,399854,5198102,67575235];; for n in [8..20] do a[n]:=12*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -78*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Sep 15 2019

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^7)/(1-13*t+90*t^7-78*t^8) )); // G. C. Greubel, Sep 15 2019

seq(coeff(series((1+t)*(1-t^7)/(1-13*t+90*t^7-78*t^8), t, n+1), t, n), n = 0..20); # G. C. Greubel, Sep 15 2019

CoefficientList[Series[(1+t)*(1-t^7)/(1-13*t+90*t^7-78*t^8), {t,0,20}],t] (* G. C. Greubel, Aug 10 2017 *)
coxG[{7, 78, -12}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 15 2019 *)

my(t='t+O('t^20)); Vec((1+t)*(1-t^7)/(1-13*t+90*t^7-78*t^8)) \\ G. C. Greubel, Aug 10 2017

def A164618_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-13*t+90*t^7-78*t^8)).list()
A164618_list(20) # G. C. Greubel, Sep 15 2019

G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(78*t^7 - 12*t^6 - 12*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).

a(n) = -78*a(n-7) + 12*Sum_{k=1..6} a(n-k). - Wesley Ivan Hurt, May 11 2021

A165874 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

Original entry on oeis.org

1, 14, 182, 2366, 30758, 399854, 5198102, 67575326, 878479238, 11420230094, 148462991131, 1930018883520, 25090245470472, 326173190917392, 4240251479342424, 55123269197863776, 716602499135588520

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

nonn

The initial terms coincide with those of A170733, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (12,12,12,12,12,12,12,12,12,-78).

a:=[14, 182, 2366, 30758, 399854, 5198102, 67575326, 878479238, 11420230094, 148462991131];; for n in [11..20] do a[n]:=12*Sum([1..9], j-> a[n-j]) -78*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 23 2019

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^10)/(1-13*t+90*t^10-78*t^11) )); // G. C. Greubel, Sep 23 2019

seq(coeff(series((1+t)*(1-t^10)/(1-13*t+90*t^10-78*t^11), t, n+1), t, n), n = 0..20); # G. C. Greubel, Sep 23 2019

CoefficientList[Series[(1+t)*(1-t^10)/(1-13*t+90*t^10-78*t^11), {t, 0, 25}], t] (* G. C. Greubel, Apr 17 2016 *)
coxG[{10, 78, -12}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 23 2019 *)

my(t='t+O('t^20)); Vec((1+t)*(1-t^10)/(1-13*t+90*t^10-78*t^11)) \\ G. C. Greubel, Sep 23 2019

def A165874_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-13*t+90*t^10-78*t^11)).list()
A165874_list(20) # G. C. Greubel, Sep 23 2019

G.f.: (t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(78*t^10 - 12*t^9 - 12*t^8 - 12*t^7 - 12*t^6 - 12*t^5 - 12*t^4 - 12*t^3 - 12*t^2 - 12*t + 1).

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A003945 Expansion of g.f. (1+x)/(1-2*x).

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A170734 Expansion of g.f.: (1+x)/(1-14*x).

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A158497 Triangle T(n,k) formed by the coordination sequences and the number of leaves for trees.

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A162783 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

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A166379 Number of reduced words of length n in Coxeter group on 14 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

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