A003948 -id:A003948 - cp's OEIS frontend

A014829 a(1)=1, a(n) = 6*a(n-1) + n.

1, 8, 51, 310, 1865, 11196, 67183, 403106, 2418645, 14511880, 87071291, 522427758, 3134566561, 18807399380, 112844396295, 677066377786, 4062398266733, 24374389600416, 146246337602515, 877478025615110, 5264868153690681, 31589208922144108, 189535253532864671, 1137211521197188050

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See p. 18.
László Tóth, On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers, arXiv:2002.06584 [math.NT], 2020. See also Proc. Amer. Math. Soc. 148 (2020), pp. 461-469.
Index entries for linear recurrences with constant coefficients, signature (8,-13,6).

[(6^(n+1)-5*n-6)/25: n in [1..30]]; // Vincenzo Librandi, Aug 23 2011

a:=n->1/5*sum(6^j-1,j=1..n): seq(a(n),n=1..20); # Zerinvary Lajos, Jun 27 2007

Join[{a=1,b=8},Table[c=7*b-6*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2011 *)
nxt[{n_,a_}]:={n+1,6a+n+1}; NestList[nxt,{1,1},30][[All,2]] (* Harvey P. Dale, Feb 12 2023 *)

Vec(x/((1 - x)^2*(1 - 6*x)) + O(x^25)) \\ Colin Barker, Jun 03 2020

a(n) = (6^(n+1) - 5*n - 6)/25. - Rolf Pleisch, Oct 25 2010
Binomial transform of x*(1+x)/(1-5*x), or A003948 with a leading 0. a(n) = Sum_{k=0..n} (n-k)*6^k = Sum_{k=0..n} k*6^(n-k); a(n) = Sum_{k=0..n} binomial(n+2,k+2)*5^k [Offset 0]. - Paul Barry, Jul 30 2004
From Colin Barker, Jun 03 2020: (Start)
G.f.: x/((1 - x)^2*(1 - 6*x)).
a(n) = 8*a(n-1) - 13*a(n-2) + 6*a(n-3) for n > 3. (End)
E.g.f.: exp(x)*(6*exp(5*x) - 5*x - 6)/25. - Elmo R. Oliveira, Mar 29 2025

A080961 Fourth binomial transform of A010686 (period 2: repeat 1,5).

Original entry on oeis.org

1, 9, 57, 321, 1713, 8889, 45417, 230001, 1158753, 5820009, 29178777, 146130081, 731358993, 3658920729, 18300980937, 91524036561, 457677578433, 2288560079049, 11443316955897, 57218134461441, 286095321353073, 1430490553902969, 7152494610927657, 35762598578876721

Offset: 0

Paul Barry, Mar 03 2003

			G.f. = 1 + 9*x + 57*x^2 + 321*x^3 + 1713*x^4 + 8889*x^5 + 45417*x^6 + 230001*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-15).

Cf. A003948, A005059, A010686, A080960, A080962.

binomtf:=func< V | [ &+[ Binomial(i-1, k-1)*V[k]: k in [1..i] ]: i in [1..#V] ] >;
binomtf(binomtf(binomtf(binomtf(&cat[ [1, 5]: n in [1..11] ])))); // Klaus Brockhaus, Nov 26 2009

[3*5^n - 2*3^n: n in [0..30]]; // Vincenzo Librandi, Dec 07 2012

A080961:=n->3*5^n-2*3^n: seq(A080961(n), n=0..30); # Wesley Ivan Hurt, Dec 08 2016

CoefficientList[Series[(1 + x)/((1 - 3*x) * (1 - 5*x)), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 07 2012 *)

a(n) = 5*a(n-1) + 4*3^(n-1).
a(n) = 3*5^n - 2*3^n.
G.f.: (1+x)/((1-3*x)*(1-5*x)). - Klaus Brockhaus, Nov 26 2009
From Mario C. Enriquez, Dec 08 2016: (Start)
a(n) = A005059(n+1) + A005059(n) = (5^(n+1)+5^n-3^(n+1)-3^n)/2.
a(n) = Sum_{k=0..n} A003948(n-k)*3^k = Sum_{k=0..n} (3^k * ceiling(Sum_{v=0..n-k} (5^v - 5^(v-2)))). (End)
a(n) = 8*a(n-1) - 15*a(n-2) for n > 1. - Wesley Ivan Hurt, Dec 08 2016
E.g.f.: exp(3*x)*(3*exp(2*x) - 2). - Stefano Spezia, Jul 23 2024

Definition corrected, edited by Klaus Brockhaus, Nov 26 2009

A208345 Triangle of coefficients of polynomials v(n,x) jointly generated with A208344; see the Formula section.

Original entry on oeis.org

1, 0, 3, 0, 1, 7, 0, 1, 3, 17, 0, 1, 3, 10, 41, 0, 1, 3, 11, 30, 99, 0, 1, 3, 12, 35, 87, 239, 0, 1, 3, 13, 40, 108, 245, 577, 0, 1, 3, 14, 45, 130, 322, 676, 1393, 0, 1, 3, 15, 50, 153, 406, 938, 1836, 3363, 0, 1, 3, 16, 55, 177, 497, 1236, 2682, 4925, 8119, 0, 1

Offset: 1

Clark Kimberling, Feb 25 2012

Row sums, u(n,1): (1,2,5,13,...), odd-indexed Fibonacci numbers.
Row sums, v(n,1): (1,3,8,21,...), even-indexed Fibonacci numbers.
As triangle T(n,k) with 0<=k<=n, it is (0, 1/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012

			First five rows:
  1
  0   3
  0   1   7
  0   1   3   17
  0   1   3   10   41
First five polynomials u(n,x):
  1, 3*x, x + 7*x^2, x + 3*x^2 + 17*x^3, x + 3*x^2 + 10*x^3 + 41*x^4.

Cf. A208344.

Cf. A084938, A000045, A000244, A001906.

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208344 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208345 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]
Table[v[n, x] /. x -> 1, {n, 1, z}]

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = x*u(n-1,x) + 2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Feb 26 2012: (Start)
As triangle T(n,k), 0<=k<=n:
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-2) - 2*T(n-2,k-1) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 3, T(n,k) = 0 if k<0 or if k>n.
G.f.: (1+(y-1)*x)/(1-(1+2*y)*x+y*(2-y)*x^2).
Sum_{k=0..n} T(n,k)*x^k = A152167(n), A000007(n), A001906(n+1), A003948(n) for x = -1, 0, 1, 2 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A078057(n), A001906(n+1), A000244(n), A081567(n), A083878(n), A165310(n) for x = 0, 1, 2, 3, 4, 5 respectively. (End)

A208509 Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.

Original entry on oeis.org

1, 3, 5, 1, 7, 5, 9, 14, 1, 11, 30, 7, 13, 55, 27, 1, 15, 91, 77, 9, 17, 140, 182, 44, 1, 19, 204, 378, 156, 11, 21, 285, 714, 450, 65, 1, 23, 385, 1254, 1122, 275, 13, 25, 506, 2079, 2508, 935, 90, 1, 27, 650, 3289, 5148, 2717, 442, 15, 29, 819, 5005, 9867

Offset: 1

Clark Kimberling, Feb 27 2012

			First five rows:
  1
  3
  5    1
  7    5
  9   14   1
First five polynomials v(n,x):
  1
  3
  5 +   x
  7 +  5x
  9 + 14x + x^2

Columns 0 to 5: A005408, A000330, A005585, A050486, A054333, A057788.
Row sums, v(n,1): A003948.
Alternating row sums, v(n,-1): A090131.
Cf. A208508.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A208508 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208509 *)

u(n,x) = u(n-1,x) + x*v(n-1,x), v(n,x) = u(n-1,x) + v(n-1,x) + 1, with u(1,x)=1, v(1,x)=1.

Conjecture: T(n,k) = binomial(n-1,2*k+1) + binomial(n,2*k+1). - Knud Werner, Jan 11 2022

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

A163317 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

Original entry on oeis.org

1, 6, 30, 150, 750, 3735, 18600, 92640, 461400, 2298000, 11445210, 57003000, 283904040, 1413987000, 7042377000, 35074632060, 174689570400, 870043225440, 4333259349600, 21581843340000, 107488595621160, 535348070440800

Offset: 0

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

nonn

The initial terms coincide with those of A003948, 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..1000
Index entries for linear recurrences with constant coefficients, signature (4, 4, 4, 4, -10).

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-5*x+14*x^5-10*x^6) )); // G. C. Greubel, May 12 2019

CoefficientList[Series[(1+x)*(1-x^5)/(1-5*x+14*x^5-10*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{4,4,4,4,-10}, {1,6,30,150,750,3735}, 30] (* G. C. Greubel, Dec 18 2016 *)
coxG[{5, 10, -4}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 12 2019 *)

my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-5*x+14*x^5-10*x^6)) \\ G. C. Greubel, Dec 18 2016

((1+x)*(1-x^5)/(1-5*x+14*x^5-10*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)/(10*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1).

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

A163922 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

Original entry on oeis.org

1, 6, 30, 150, 750, 3750, 18735, 93600, 467640, 2336400, 11673000, 58320000, 291375210, 1455753000, 7273154040, 36337737000, 181548627000, 907043385000, 4531720872060, 22641137570400, 113118421225440, 565156109349600

Offset: 0

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

nonn

The initial terms coincide with those of A003948, 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..1000
Index entries for linear recurrences with constant coefficients, signature (4,4,4,4,4,-10).

a:=[6,30,150,750,3750,18735];; for n in [7..30] do a[n]:=4*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -10*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 10 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-5*t+14*t^6-10*t^7) )); // G. C. Greubel, Aug 10 2019

seq(coeff(series((1+t)*(1-t^6)/(1-5*t+14*t^6-10*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 10 2019

coxG[{6,10,-4,30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 13 2017 *)
CoefficientList[Series[(1+t)*(1-t^6)/(1-5*t+14*t^6-10*t^7), {t,0,30}], t] (* G. C. Greubel, Aug 07 2017 *)

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-5*t+14*t^6-10*t^7)) \\ G. C. Greubel, Aug 07 2017

def A163922_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-5*t+14*t^6-10*t^7)).list()
A163922_list(30) # G. C. Greubel, Aug 10 2019

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(10*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1).

a(n) = -10*a(n-6) + 4*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021

A164365 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

Original entry on oeis.org

1, 6, 30, 150, 750, 3750, 18750, 93735, 468600, 2342640, 11711400, 58548000, 292695000, 1463250000, 7315125210, 36570003000, 182821904040, 913968987000, 4569142377000, 22842199635000, 114193439625000, 570879418872060

Offset: 0

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

nonn

The initial terms coincide with those of A003948, 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..1000
Index entries for linear recurrences with constant coefficients, signature (4,4,4,4,4,4,-10).

a:=[6, 30, 150, 750, 3750, 18750, 93735];; for n in [8..30] do a[n]:=4*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -10*a[n-7]; od; Concatenation([1], a); # G. C. Greubel, Aug 28 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^7)/(1-5*t+14*t^7-10*t^8) )); // G. C. Greubel, Aug 28 2019

seq(coeff(series((1+t)*(1-t^7)/(1-5*t+14*t^7-10*t^8), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 28 2019

CoefficientList[Series[(1+t)*(1-t^7)/(1-5*t+14*t^7-10*t^8), {t, 0, 30}], t] (* G. C. Greubel, Sep 15 2017 *)
coxG[{7, 10, -4}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 28 2019 *)

my(t='t+O('t^30)); Vec((1+t)*(1-t^7)/(1-5*t+14*t^7-10*t^8)) \\ G. C. Greubel, Sep 15 2017

def A164365_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-5*t+14*t^7-10*t^8)).list()
A164365_list(30) # G. C. Greubel, Aug 28 2019

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

a(n) = -10*a(n-7) + 4*Sum_{k=1..6} a(n-k). - Wesley Ivan Hurt, May 11 2021

A165777 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

Original entry on oeis.org

1, 6, 30, 150, 750, 3750, 18750, 93750, 468750, 2343750, 11718735, 58593600, 292967640, 1464836400, 7324173000, 36620820000, 183103875000, 915518250000, 4577585625000, 22887900000000, 114439359375210, 572196093753000, 2860976953154040, 14304867187737000

Offset: 0

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

nonn

The initial terms coincide with those of A003948, 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 (4,4,4,4,4,4,4,4,4,-10).

Cf. A003948, A154638.

a:=[6, 30, 150, 750, 3750, 18750, 93750, 468750, 2343750, 11718735];; for n in [11..30] do a[n]:=4*Sum([1..9], j-> a[n-j]) -10*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 17 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^10)/(1-5*t+14*t^10-10*t^11) )); // G. C. Greubel, Sep 17 2019

A165777 := proc(n)
coeftayl( (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)/(10*t^10 - 4*t^9 - 4*t^8 - 4*t^7 - 4*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1), t=0, n);
end proc:
seq(A165777(n), n=0..25); # Wesley Ivan Hurt, Nov 14 2014

CoefficientList[Series[(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)/(10*t^10 - 4*t^9 - 4*t^8 - 4*t^7 - 4*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1), {t, 0, 25}], t] (* Wesley Ivan Hurt, Nov 14 2014 *)
coxG[{10, 10, -4}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 17 2019 *)

my(t='t+O('t^30)); Vec((1+t)*(1-t^10)/(1-5*t+14*t^10-10*t^11)) \\ G. C. Greubel, Sep 17 2019

def A165777_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-5*t+14*t^10-10*t^11)).list()
A165777_list(30) # G. C. Greubel, Sep 17 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)/(10*t^10 - 4*t^9 - 4*t^8 - 4*t^7 - 4*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1).

More terms from Wesley Ivan Hurt, Nov 14 2014

A166364 Number of reduced words of length n in Coxeter group on 6 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

Original entry on oeis.org

1, 6, 30, 150, 750, 3750, 18750, 93750, 468750, 2343750, 11718750, 58593735, 292968600, 1464842640, 7324211400, 36621048000, 183105195000, 915525750000, 4577627625000, 22888132500000, 114440634375000, 572203031250000

Offset: 0

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

nonn

The initial terms coincide with those of A003948, 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 (4,4,4,4,4,4,4,4,4,4,-10).

seq(coeff(series((1+t)*(1-t^11)/(1-5*t+14*t^11-10*t^12), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Mar 13 2020

CoefficientList[Series[(1+t)*(1-t^11)/(1-5*t+14*t^11-10*t^12), {t,0,30}], t] (* G. C. Greubel, May 10 2016 *)
coxG[{11,10,-4,30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jul 13 2016 *)

def A166364_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^11)/(1-5*t+14*t^11-10*t^12) ).list()
A166364_list(30) # G. C. Greubel, Mar 13 2020

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)/(10*t^11 - 4*t^10 - 4*t^9 - 4*t^8 - 4*t^7 - 4*t^6 - 4*t^5 - 4*t^4 - 4*t^3 - 4*t^2 - 4*t + 1).

cp's OEIS Frontend

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