A003949 -id:A003949 - cp's OEIS frontend

A108982 Inverse binomial of A003949.

1, 6, 29, 146, 729, 3646, 18229, 91146, 455729, 2278646, 11393229, 56966146, 284830729, 1424153646, 7120768229, 35603841146, 178019205729, 890096028646, 4450480143229, 22252400716146, 111262003580729, 556310017903646

Offset: 0

Philippe Deléham, Jul 23 2005

nonn

Let A be the Hessenberg matrix of order n, defined by: A[1,j] = 1, A[i,i] = -3, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n>=1, a(n-1) = charpoly(A,2). - Milan Janjic, Jan 27 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4, 5).

List([0..25], n-> (7*5^n - (-1)^n)/6); # G. C. Greubel, Sep 25 2019

[(7*5^n - (-1)^n)/6: n in [0..25]]; // G. C. Greubel, Sep 25 2019

seq((7*5^n - (-1)^n)/6, n=0..25); # G. C. Greubel, Sep 25 2019

LinearRecurrence[{4,5},{1,6},30] (* Harvey P. Dale, Jul 07 2013 *)

vector(26, n, (7*5^(n-1) + (-1)^n)/6) \\ G. C. Greubel, Sep 25 2019

[(7*5^n - (-1)^n)/6 for n in (0..25)] # G. C. Greubel, Sep 25 2019

a(n) = 4*a(n-1) + 5*a(n-2), a(0) = 1, a(1) = 6.
a(2n) = 5*a(2n-1) - 1 and a(2n+1) = 5*a(2n) + 1.
From G. C. Greubel, Sep 25 2019: (Start)
G.f.: (1 + 2*x)/((1+x)*(1-5*x)).
E.g.f.: (7*exp(5*x) - exp(-x))/6. (End)

Corrected and extended by T. D. Noe, Nov 07 2006

A169452 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^33 = I.

Original entry on oeis.org

1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, 11757312, 70543872, 423263232, 2539579392, 15237476352, 91424858112, 548549148672, 3291294892032, 19747769352192, 118486616113152, 710919696678912, 4265518180073472

Offset: 0

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

The initial terms coincide with those of A003949, although the two sequences are eventually different.

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, -15).

R:=PowerSeriesRing(Integers(), 25); Coefficients(R!( (1+x)*(1-x^3)/(1-14*x+104*x^3-91*x^4) )); // G. C. Greubel, May 01 2019

gf:= (t+1) *(t^2+t+1) *(t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1) *(t^20-t^19+t^17-t^16 +t^14-t^13+t^11-t^10+t^9-t^7+t^6-t^4+t^3- t+1) / (15*t^33-5*t^32-5*t^31-5*t^30-5*t^29 -5*t^28-5*t^27 -5*t^26-5*t^25 -5*t^24 -5*t^23-5*t^22-5*t^21-5*t^20 -5*t^19-5*t^18-5*t^17 -5*t^16 -5*t^15 -5*t^14-5*t^13-5*t^12-5*t^11-5*t^10-5*t^9-5*t^8-5*t^7 -5*t^6 -5*t^5-5*t^4 -5*t^3-5*t^2-5*t+1):
S:= series(gf,t,101):
seq(coeff(S,t,j),j=0..100); # Robert Israel, Aug 26 2014

coxG[{pwr_,c1_,c2_,trms_:20}]:=Module[{num=Total[2t^Range[pwr-1]]+t^pwr+ 1, den =Total[c2*t^Range[pwr-1]]+c1*t^pwr+1},CoefficientList[ Series[ num/den,{t,0,trms}],t]]; coxG[{33,15,-5,30}]
(* "pwr" is the largest exponent in the g.f.;
"c1" is the first coefficient in the denominator of the g.f.;
"c2" is the second coefficient in the denominator of the g.f.;
"trms" is the number of terms desired (with a default number of 20) *)
(* Harvey P. Dale, Aug 16 2014 *)
CoefficientList[Series[(1+x)*(1-x^33)/(1-6*x+20*x^33-15*x^34), {x,0,25}], x] (* G. C. Greubel, May 01 2019 *)

my(x='x+O('x^25)); Vec((1+x)*(1-x^33)/(1-6*x+20*x^33-15*x^34)) \\ G. C. Greubel, May 01 2019

((1+x)*(1-x^33)/(1-6*x+20*x^33-15*x^34)).series(x, 25).coefficients(x, sparse=False) # G. C. Greubel, May 01 2019

G.f.: (t^33 + 2*t^32 + 2*t^31 + 2*t^30 + 2*t^29 + 2*t^28 + 2*t^27 + 2*t^26 + 2*t^25 + 2*t^24 + 2*t^23 + 2*t^22 + 2*t^21 + 2*t^20 + 2*t^19 + 2*t^18 + 2*t^17 + 2*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)/(15*t^33 - 5*t^32 - 5*t^31 - 5*t^30 - 5*t^29 - 5*t^28 - 5*t^27 - 5*t^26 - 5*t^25 - 5*t^24 - 5*t^23 - 5*t^22 - 5*t^21 - 5*t^20 - 5*t^19 - 5*t^18 - 5*t^17 - 5*t^16 - 5*t^15 - 5*t^14 - 5*t^13 - 5*t^12 - 5*t^11 - 5*t^10 - 5*t^9 - 5*t^8 - 5*t^7 - 5*t^6 - 5*t^5 - 5*t^4 - 5*t^3 - 5*t^2 - 5*t + 1).
G.f.: (1+x)*(1-x^33)/(1 - 6*x + 20*x^33 - 15*x^34). - G. C. Greubel, May 01 2019
a(n) = -15*a(n-33) + 5*Sum_{k=1..32} a(n-k). - Wesley Ivan Hurt, May 06 2021

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

Original entry on oeis.org

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

A003947 Expansion of (1+x)/(1-4*x).

Original entry on oeis.org

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971520, 83886080, 335544320, 1342177280, 5368709120, 21474836480, 85899345920, 343597383680, 1374389534720, 5497558138880, 21990232555520, 87960930222080, 351843720888320

Offset: 0

N. J. A. Sloane

Coordination sequence for infinite tree with valency 5.
For n>=1, a(n+1) is equal to the number of functions f:{1,2,...,n+1}->{1,2,3,4,5} such that for fixed, different x_1, x_2,...,x_n in {1,2,...,n+1} and fixed y_1, y_2,...,y_n in {1,2,3,4,5} we have f(x_i)<>y_i, (i=1,2,...,n). - Milan Janjic, May 10 2007
Number of length-n strings of 5 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
Create a rectangular prism with edges of lengths 2^(n-2), 2^(n-1), and 2^(n) starting at n=2; then the surface area = a(n). - J. M. Bergot, Aug 08 2013

T. D. Noe, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 306
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (4).
Index entries for sequences related to trees

Cf. A003948, A003949. Column 5 in A265583.

Concatenation([1], List([1..30], n-> 5*4^(n-1) )); # G. C. Greubel, Aug 10 2019

[1] cat [5*4^(n-1): n in [1..30]]; // G. C. Greubel, Aug 10 2019

k := 5; if n = 0 then 1 else k*(k-1)^(n-1); fi;

q = 5; Join[{a = 1}, Table[If[n != 0, a = q*a - a, a = q*a], {n, 0, 25}]] (* and *) Join[{1}, 5*4^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
LinearRecurrence[{4},{1,5},30] (* Harvey P. Dale, Apr 19 2015 *)

a(n)=5*4^n\4 \\ Charles R Greathouse IV, Sep 08 2011

[1]+[5*4^(n-1) for n in (1..30)] # G. C. Greubel, Aug 10 2019

Binomial transform of A060925. Its binomial transform is A003463 (without leading zero). - Paul Barry, May 19 2003
From Paul Barry, May 19 2003: (Start)
a(n) = (5*4^n - 0^n)/4.
G.f.: (1+x)/(1-4*x).
E.g.f.: (5*exp(4*x) - exp(0))/4. (End)
a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 3. - Philippe Deléham, Jul 10 2005
a(n) = A146523(n)*A011782(n). - R. J. Mathar, Jul 08 2009
a(n) = 5*A000302(n-1), n>0.
a(n) = 4*a(n-1), n>1. - Vincenzo Librandi, Dec 31 2010
G.f.: 2+x- 2/G(0), where G(k)= 1 + 1/(1 - x*(5*k-4)/(x*(5*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 04 2013

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

A003948 Expansion of (1+x)/(1-5*x).

Original entry on oeis.org

1, 6, 30, 150, 750, 3750, 18750, 93750, 468750, 2343750, 11718750, 58593750, 292968750, 1464843750, 7324218750, 36621093750, 183105468750, 915527343750, 4577636718750, 22888183593750, 114440917968750, 572204589843750, 2861022949218750, 14305114746093750

Offset: 0

N. J. A. Sloane

Coordination sequence for infinite tree with valency 6.
The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954, m is 2, 3, 4, 5, 6. - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001
Hamiltonian path in S_4 X P_2n.
For n>=1, a(n+1) is equal to the number of functions f:{1,2,...,n+1}->{1,2,3,4,5,6} such that for fixed, different x_1, x_2,...,x_n in {1,2,...,n+1} and fixed y_1, y_2,...,y_n in {1,2,3,4,5,6} we have f(x_i)<>y_i, (i=1..n). - Milan Janjic, May 10 2007
For n>=1, a(n) equals the numbers of words of length n over the alphabet {0..5} with no two adjacent letters identical.  - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, May 30 2017]
a(n) equals the numbers of sequences of length n on {0,...,5} where no two adjacent terms differ by three. - David Nacin, May 30 2017
It appears that these are the only n>1 for which alpha(n)=2n, where alpha(n) is the entry point of n in the Fibonacci sequence, see A001177.  - Philippe Schnoebelen, Apr 11 2024

T. D. Noe, Table of n, a(n) for n = 0..200
F. Faase, Counting Hamiltonian cycles in product graphs
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 307
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
A. M. Nemirovsky et al., Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (5).
Index entries for sequences related to trees

Cf. A003946, A003947, A003949, A003950, A003952, A003954, A029653.

Concatenation([1], List([1..30], n-> 6*5^(n-1) )); # G. C. Greubel, Sep 24 2019

[1] cat [6*5^(n-1): n in [1..30]]; // G. C. Greubel, Sep 24 2019

k := 6; if n = 0 then 1 else k*(k-1)^(n-1); fi;

q = 6; Join[{a = 1}, Table[If[n != 0, a = q*a - a, a = q*a], {n, 0, 25}]] (* and *) Join[{1}, 6*5^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
Join[{1},NestList[5#&,6,30]] (* Harvey P. Dale, Dec 31 2013 *)
CoefficientList[Series[(1+x)/(1-5x), {x,0,30}], x] (* Michael De Vlieger, Dec 10 2016 *)

Vec((1+x)/(1-5*x)+O(x^30)) \\ Charles R Greathouse IV, Nov 20 2012

[1]+[6*5^(n-1) for n in (1..30)] # G. C. Greubel, Sep 24 2019

G.f.: (1+x)/(1-5*x).
a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 4. - Philippe Deléham, Jul 10 2005
The Hankel transform of this sequence is [1,-6,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007
a(n) = 6*5^(n-1) for n>0, a(0)=1. - Vincenzo Librandi, Nov 18 2010
G.f.: 2/x - 5 - 8/(x*U(0)) where U(k)= 1 + 2/(3^k - 3^k/(2 + 1 - 12*x*3^k/(6*x*3^k + 1/U(k+1)))) ; (continued fraction, 4-step). - Sergei N. Gladkovskii, Oct 30 2012
E.g.f.: (6*exp(5*x) - 1)/5. - Ilya Gutkovskiy, Dec 10 2016
Sum_{n>=0} 1/a(n) = 29/24. - Bernard Schott, Oct 25 2021

Definition corrected by Frans J. Faase, Feb 07 2009

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

A003950 Expansion of g.f.: (1+x)/(1-7*x).

Original entry on oeis.org

1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828856, 2259801992, 15818613944, 110730297608, 775112083256, 5425784582792, 37980492079544, 265863444556808, 1861044111897656, 13027308783283592, 91191161482985144, 638338130380896008

Offset: 0

N. J. A. Sloane

Coordination sequence for infinite tree with valency 8.
The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954 m is 2, 3, 4, 5, 6 . - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001.
For n>=1, a(n) equals the number of words of length n on the alphabet {0,1,...,7} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, May 31 2017]
a(n) is the number of octonary sequences of length n such that no two consecutive terms have distance 4. - David Nacin, May 31 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 309
A. M. Nemirovsky et al., Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (7).
Index entries for sequences related to trees

Cf. A003948, A003949, A003951.

k:=8;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

[1] cat [8*7^(n-1): n in [1..25]]; // Vincenzo Librandi, Dec 11 2012

k:=8; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # modified by G. C. Greubel, Sep 24 2019

Join[{1}, 8*7^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
CoefficientList[Series[(1+x)/(1-7*x), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 10 2012 *)

a(n)=if(n,8*7^(n-1),1) \\ Charles R Greathouse IV, Mar 22 2016

k=8; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Sep 24 2019

a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 6. - Philippe Deléham, Jul 10 2005
From Philippe Deléham, Nov 21 2007: (Start)
a(n) = 8*7^(n-1) for n>=1, a(0)=1 .
G.f.: (1+x)/(1-7x).
The Hankel transform of this sequence is [1,-8,0,0,0,0,0,0,0,0,...]. (End)
a(0)=1, a(1)=8, a(n) = 7*a(n-1). - Vincenzo Librandi, Dec 10 2012
E.g.f.: (8*exp(7*x) - 1)/7. - G. C. Greubel, Sep 24 2019

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

A003952 Expansion of g.f.: (1+x)/(1-9*x).

Original entry on oeis.org

1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, 430467210, 3874204890, 34867844010, 313810596090, 2824295364810, 25418658283290, 228767924549610, 2058911320946490, 18530201888518410, 166771816996665690, 1500946352969991210, 13508517176729920890

Offset: 0

N. J. A. Sloane, Mar 15 1996

Coordination sequence for infinite tree with valency 10.
The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954 m is 2, 3, 4, 5, 6 . - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001
Except 1, all terms are in A033583. - Vincenzo Librandi, May 26 2014
For n>=1, a(n) equals the number of words of length n on alphabet {0,1,...,9} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, May 31 2017]
a(n) is the number of sequences over the alphabet {0,1,...,9} of length n such that no two consecutive terms have distance 5. - David Nacin, May 31 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 311
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (9).
Index entries for sequences related to trees

Cf. A003949, A003950, A003951.

k:=10;; Concatenation([1], List([1..20], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

[(10*9^n-0^n)/9: n in [0..20] ]; // Vincenzo Librandi, Aug 19 2011

k:= 10; seq(`if`(n = 0, 1, k*(k-1)^(n-1)), n = 0..20); # modified by G. C. Greubel, Sep 24 2019

Join[{1}, 10*9^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
Join[{1},NestList[9#&,10,20]] (* Harvey P. Dale, Sep 01 2021 *)

a(n)=10*9^n\9 \\ Charles R Greathouse IV, Sep 08 2011

k=10; [1]+[k*(k-1)^(n-1) for n in (1..20)] # G. C. Greubel, Sep 24 2019

a(n) = (10*9^n - 0^n)/9. Binomial transform is A000042. - Paul Barry, Jan 29 2004
G.f.: (1+x)/(1-9*x). - Philippe Deléham, Jan 31 2004
a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 8. - Philippe Deléham, Jul 10 2005
The Hankel transform of this sequence is: [1,-10,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007
E.g.f.: (10*exp(9*x) - 1)/9. - G. C. Greubel, Sep 24 2019

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

A151791 a(1)=1; for n > 1, a(n) = 7*6^(wt(n-1)-1).

Original entry on oeis.org

1, 7, 7, 42, 7, 42, 42, 252, 7, 42, 42, 252, 42, 252, 252, 1512, 7, 42, 42, 252, 42, 252, 252, 1512, 42, 252, 252, 1512, 252, 1512, 1512, 9072, 7, 42, 42, 252, 42, 252, 252, 1512, 42, 252, 252, 1512, 252, 1512, 1512, 9072, 42, 252, 252, 1512, 252, 1512, 1512, 9072, 252

Offset: 1

N. J. A. Sloane, Jun 25 2009

wt(n) is the Hamming weight = binary weight of n (A000120).

			From _Omar E. Pol_, Feb 26 2015: (Start)
Written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
7;
7,42;
7,42,42,252;
7,42,42,252,42,252,252,1512;
7,42,42,252,42,252,252,1512,42,252,252,1512,252,1512,1512,9072;
7,42,42,252,42,252,252,1512,42,252,252,1512,252,1512,1512,9072,42,252,252,1512,252,1512,1512,9072,252,...
It appears that the right border gives A003949.
It appears that the row sums give A000420.
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..8192

Cf. A011782, A000120, A000420, A151792 (partial sums).

a[n_] := 7*6^(Total@ IntegerDigits[n - 1, 2] - 1); a[1] = 1; Array[a, 57] (* Michael De Vlieger, Nov 01 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

A164369 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

Original entry on oeis.org

1, 7, 42, 252, 1512, 9072, 54432, 326571, 1959300, 11755065, 70525980, 423129420, 2538617760, 15230754000, 91378809060, 548238566925, 3289225689750, 19734119944875, 118397314970550, 710339464409400, 4261770250642800

Offset: 0

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

nonn

The initial terms coincide with those of A003949, 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 (5, 5, 5, 5, 5, 5, -15).

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^7)/(1-6*x+20*x^7-15*x^8) )); // G. C. Greubel, Apr 25 2019

CoefficientList[Series[(1+x)*(1-x^7)/(1-6*x+20*x^7-15*x^8), {x, 0, 30}], x] (* G. C. Greubel, Sep 17 2017, modified Apr 25 2019 *)
coxG[{7, 15, -5, 30}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 25 2019 *)

my(x='x+O('x^30)); Vec((1+x)*(1-x^7)/(1-6*x+20*x^7-15*x^8)) \\ G. C. Greubel, Sep 17 2017, modified Apr 25 2019

((1+x)*(1-x^7)/(1-6*x+20*x^7-15*x^8)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 25 2019

G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(15*t^7 - 5*t^6 - 5*t^5 - 5*t^4 - 5*t^3 - 5*t^2 - 5*t + 1).

G.f.: (1+x)*(1-x^7)/(1 -6*x +20*x^7 -15*x^8). - G. C. Greubel, Apr 25 2019

cp's OEIS Frontend

A108982 Inverse binomial of A003949.

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A169452 Number of reduced words of length n in Coxeter group on 7 generators S_i with relations (S_i)^2 = (S_i S_j)^33 = I.

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

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A003947 Expansion of (1+x)/(1-4*x).

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A003948 Expansion of (1+x)/(1-5*x).

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A003950 Expansion of g.f.: (1+x)/(1-7*x).

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