A005021 -id:A005021 - cp's OEIS frontend

A216054 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 1 or if k-n >= 6, T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 5, 0, 0, 0, 0, 5, 9, 5, 0, 0, 0, 0, 5, 14, 14, 0, 0, 0, 0, 0, 0, 19, 28, 14, 0, 0, 0, 0, 0, 0, 19, 47, 42, 0, 0, 0, 0, 0, 0, 0, 0, 66, 89, 42, 0, 0, 0, 0, 0, 0, 0, 0, 66, 155, 131, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 221, 286, 131, 0, 0

Offset: 0

Philippe Deléham, Mar 16 2013

A hexagon arithmetic of E. Lucas.

			Square array begins:
1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=0
0, 1, 2, 3, 4, 5, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=1
0, 0, 2, 5, 9, 14, 19, 19, 0, 0, 0, 0, 0, 0, 0, ... row n=2
0, 0, 0, 5, 14, 28, 47, 66, 66, 0, 0, 0, 0, 0, 0, ... row n=3
0, 0, 0, 0, 14, 42, 89, 155, 221, 221, 0, 0, 0, 0, ... row n=4
0, 0, 0, 0, 0, 0, 42, 131, 286, 507, 728, 728, 0, 0, ... row n=5
0, 0, 0, 0, 0, 0, 131, 417, 924, 1652, 2380, 2380, 0, ... row n=6
...

E. Lucas, Théorie des nombres, A.Blanchard, Paris, 1958, Tome 1, p.89

Cf. A006053, A052547, A096976, A187066,

Cf. Similar sequences A216230, A216228, A216226, A216238

Clear[t]; t[0, k_ /; k <= 5] = 1; t[n_, k_] /; k < n || k > n+5 = 0; t[n_, k_] := t[n, k] = t[n-1, k] + t[n, k-1]; Table[t[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 18 2013 *)

T(n,n) = A080937(n).
T(n,n+1) = A080937(n+1).
T(n,n+2) = A094790(n+1).
T(n,n+3) = A094789(n+1).
T(n,n4) = T(n,n+5) = A005021(n).
Sum_{k, 0<=k<=n} T(n-k,k) = A028495(n).

A216235 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 2 or if k-n >= 5, T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 2, 0, 1, 4, 5, 0, 0, 0, 5, 9, 5, 0, 0, 0, 5, 14, 14, 0, 0, 0, 0, 0, 19, 28, 14, 0, 0, 0, 0, 0, 19, 47, 42, 0, 0, 0, 0, 0, 0, 0, 66, 89, 42, 0, 0, 0, 0, 0, 0, 0, 66, 155, 131, 0, 0, 0, 0, 0, 0, 0, 0, 0, 221, 286, 131, 0, 0, 0, 0, 0, 0, 0, 0, 0, 221, 507, 417, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 14 2013

Arithmetic hexagon of E. Lucas.

			Square array begins:
  1, 1, 1,  1,  1,   0,   0,   0,   0,   0, ... row n=0
  1, 2, 3,  4,  5,   5,   0,   0,   0,   0, ... row n=1
  0, 2, 5,  9, 14,  19,  19,   0,   0,   0, ... row n=2
  0, 0, 5, 14, 28,  47,  66,  66,   0,   0, ... row n=3
  0, 0, 0, 14, 42,  89, 155, 221, 221,   0, ... row n=4
  0, 0, 0,  0, 42, 131, 286, 507, 728, 728, ... row n=5
  ...

E. Lucas, Théorie des nombres, Gauthier-Villars, Paris 1891, Tome 1, p. 89.

Cf. A005021, A080937, A094789, A094790.

Similar sequences: A216201, A216210, A216216, A216218, A216219, A216220, A216226, A216228, A216229, A216230, A216232.

T(n,n) = T(n+1,n) = A080937(n+1).
T(n,n+1) = A094790(n+1).
T(n,n+2) = A094789(n+1).
T(n,n+3) = T(n,n+4) = A005021(n).
Sum_{k=0..n} T(n-k,k) = A028495(n+1). - Philippe Deléham, Mar 23 2013

A005022 Number of walks of length 2n+6 in the path graph P_7 from one end to the other.

Original entry on oeis.org

6, 26, 100, 364, 1288, 4488, 15504, 53296, 182688, 625184, 2137408, 7303360, 24946816, 85196928, 290926848, 993379072, 3391793664, 11580678656, 39539651584, 134998297600, 460915984384, 1573671536640, 5372862566400, 18344123969536, 62630804299776

Offset: 1

N. J. A. Sloane, Simon Plouffe

			Example: a(1)=6 because in the path ABCDEFG we have ABABCDEFG, ABCBCDEFG, ABCDCDEFG, ABCDEDEFG, ABCDEFEFG and ABCDEFGFG. - _Emeric Deutsch_, Apr 02 2004

W. Feller, An Introduction to Probability Theory and its Applications, 3rd ed, Wiley, New York, 1968, p. 96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. J. Everett, P. R. Stein, The combinatorics of random walk with absorbing barriers, Discrete Math. 17 (1977), no. 1, 27-45.
C. J. Everett, P. R. Stein, The combinatorics of random walk with absorbing barriers, Discrete Math. 17 (1977), no. 1, 27-45. [Annotated scanned copy]
P. Flajolet, J.-C. Raoult, and J. Vuillemin, The number of registers required for evaluating arithmetic expressions, Theoret. Comput. Sci. 9 (1979), no. 1, 99-125.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Xavier Gérard Viennot, A Strahler bijection between Dyck paths and planar trees. Formal power series and algebraic combinatorics (Barcelona, 1999). Discrete Math. 246 (2002), no. 1-3, 317--329. MR1887493 (2003b:05013).
Index entries for linear recurrences with constant coefficients, signature (6,-10,4).

See A094811 for another version.

I:=[6, 26, 100]; [n le 3 select I[n] else 6*Self(n-1)-10*Self(n-2)+4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 08 2013

a:=k->sum(binomial(6+2*k,8*j+k-2),j=ceil((2-k)/8)..floor((8+k)/8))-sum(binomial(6+2*k,8*j+k-1),j=ceil((1-k)/8)..floor((7+k)/8)): seq(a(k),k=1..28);
A005022:=-1/((2*z-1)*(2*z**2-4*z+1)) -1; # [Conjectured (correctly) by Simon Plouffe in his 1992 dissertation. Gives sequence with an additional leading term of 1.]

CoefficientList[Series[-(2 (2 z^2 - 5 z + 3))/(4 z^3 - 10 z^2 + 6 z - 1), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 27 2011 *)
CoefficientList[Series[(1 / x) (1 / (1 - 6 x + 10 x^2 - 4 x^3) - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 08 2013 *)

Vec(2*(1-x)*(3-2*x) / ((1-2*x)*(1-4*x+2*x^2)) + O(x^50)) \\ Colin Barker, Apr 27 2016

G.f.: 1/(1-6x+10x^2-4x^3)-1.
a(n) = 6*a(n-1)-10*a(n-2)+4*a(n-3). - Emeric Deutsch, Apr 02 2004
a(k) = sum(binomial(6+2*k, 8*j+k-2)-binomial(6+2*k, 8*j+k-1), j=-infinity..infinity) (a finite sum).
The g.f. x^3/(1-6*x+10*x^2-4*x^3) occurs on page 320 of Viennot, 2002.
a(n) = -2^(1+n)+(3/2-sqrt(2))*(2-sqrt(2))^n+(3/2+sqrt(2))*(2+sqrt(2))^n - Colin Barker, Apr 27 2016
E.g.f.: (-2 + 3*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x))*exp(2*x) - 1. - Ilya Gutkovskiy, Apr 27 2016

Edited by Emeric Deutsch, Apr 28 2004

A005024 Number of walks of length 2n+8 in the path graph P_9 from one end to the other.

Original entry on oeis.org

8, 43, 196, 820, 3264, 12597, 47652, 177859, 657800, 2417416, 8844448, 32256553, 117378336, 426440955, 1547491404, 5610955132, 20332248992, 73645557469, 266668876540, 965384509651, 3494279574288, 12646311635088, 45764967830976

Offset: 1

N. J. A. Sloane

W. Feller, An Introduction to Probability Theory and its Applications, 3rd ed, Wiley, New York, 1968, p. 96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. J. Everett, P. R. Stein, The combinatorics of random walk with absorbing barriers, Discrete Math. 17 (1977), no. 1, 27-45.
C. J. Everett, P. R. Stein, The combinatorics of random walk with absorbing barriers, Discrete Math. 17 (1977), no. 1, 27-45. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Cf. A005023. Truncated version of A094865.

I:=[8, 43, 196, 820]; [n le 4 select I[n] else 8*Self(n-1)-21*Self(n-2)+20*Self(n-3)-5*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 08 2013

a:=k->sum(binomial(8+2*k,10*j+k-2),j=ceil((2-k)/10)..floor((10+k)/10))-sum(binomial(8+2*k,10*j+k-1),j=ceil((1-k)/10)..floor((9+k)/10)): seq(a(k),k=1..28);
A005024:=-(-8+21*z-20*z**2+5*z**3)/(5*z**2-5*z+1)/(z**2-3*z+1); # conjectured by Simon Plouffe in his 1992 dissertation

CoefficientList[Series[(-5 z^3 + 20 z^2 - 21 z + 8)/((z^2 - 3 z + 1) (5 z^2 - 5 z + 1)), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 27 2011 *)
CoefficientList[Series[(1 / x) (1 / (1 - 8 x + 21 x^2 - 20 x^3 + 5 x^4) - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 08 2013 *)

x='x+O('x^66); Vec(-1+1/((1-3*x+x^2)*(1-5*x+5*x^2))) \\ Joerg Arndt, May 01 2013

From Emeric Deutsch, Apr 02 2004: (Start)
G.f. (assuming a(0)=1): 1/(1 - 8x + 21x^2 - 20x^3 + 5x^4) - 1.
a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4). (End)
a(k) = sum(binomial(8+2k, 10j+k-2)-binomial(8+2k, 10j+k-1), j=-infinity..infinity) (a finite sum).

Better definition from Emeric Deutsch, Apr 02 2004

A005025 Random walks.

Original entry on oeis.org

9, 53, 260, 1156, 4845, 19551, 76912, 297275, 1134705, 4292145, 16128061, 60304951, 224660626, 834641671, 3094322026, 11453607152, 42344301686, 156404021389, 577291806894, 2129654436910, 7853149169635, 28949515515376, 106692395098433, 393137817645838

Offset: 1

N. J. A. Sloane

Number of walks of length 2n+9 in the path graph P_10 from one end to the other one. - Emeric Deutsch, Apr 02 2004

W. Feller, An Introduction to Probability Theory and its Applications, 3rd ed, Wiley, New York, 1968, p. 96
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. J. Everett, P. R. Stein, The combinatorics of random walk with absorbing barriers, Discrete Math. 17 (1977), no. 1, 27-45.
C. J. Everett, P. R. Stein, The combinatorics of random walk with absorbing barriers, Discrete Math. 17 (1977), no. 1, 27-45. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (9,-28,35,-15,1).

I:=[9,53,260,1156,4845]; [n le 5 select I[n] else 9*Self(n-1)-28*Self(n-2)+35*Self(n-3)-15*Self(n-4)+Self(n-5): n in [1..30]]; // Vincenzo Librandi, Jun 20 2017

a:=k->sum(binomial(9+2*k,11*j+k-2),j=ceil((2-k)/11)..floor((11+k)/11))-sum(binomial(9+2*k,11*j+k-1),j=ceil((1-k)/11)..floor((10+k)/11)): seq(a(k),k=1..28);
A005025:=-(9-28*z+35*z**2-15*z**3+z**4)/(-1+9*z-28*z**2+35*z**3-15*z**4+z**5); # Simon Plouffe in his 1992 dissertation

LinearRecurrence[{9, -28, 35, -15, 1}, {9, 53, 260, 1156, 4845}, 25] (* Vincenzo Librandi, Jun 20 2017 *)

From Emeric Deutsch, Apr 02 2004: (Start)
G.f.: 1/(1 - 9*x + 28*x^2 - 35*x^3 + 15*x^4 - x^5) - 1.
a(n) = 9*a(n-1) - 28*a(n-2) + 35*a(n-3) - 15*a(n-4) + a(n-5). (End)
a(k) = Sum_{j=-infinity..infinity} (binomial(9+2*k, 11j+k-2) - binomial(9+2*k, 11j+k-1)) (a finite sum).

A072266 Number of words of length 2n generated by the two letters s and t that reduce to the identity 1 using the relations sssssss=1, tt=1 and stst=1. The generators s and t along with the three relations generate the 14-element dihedral group D7.

Original entry on oeis.org

1, 1, 3, 10, 35, 126, 462, 1717, 6451, 24463, 93518, 360031, 1394582, 5430530, 21242341, 83411715, 328589491, 1297937234, 5138431851, 20380608990, 80960325670, 322016144629, 1282138331587, 5109310929719, 20374764059254

Offset: 0

Jamaine Paddyfoot (jay_paddyfoot(AT)hotmail.com) and John W. Layman, Jul 08 2002

			The words tttt=tsts=stst=1 so a(2)=3.

Colin Barker, Table of n, a(n) for n = 0..1000
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.
Index entries for linear recurrences with constant coefficients, signature (9,-26,25,-4).

Bisection of A377573.

LinearRecurrence[{9,-26,25,-4},{1,1,3,10,35},30] (* Harvey P. Dale, Apr 16 2022 *)

a(n)=if(n<1,n==0,sum(k=-(n-1)\7,(n-1)\7,C(2*n-1,n+7*k)))

Vec((1 - 8*x + 20*x^2 - 16*x^3 + 2*x^4) / ((1 - 4*x)*(1 - 5*x + 6*x^2 - x^3)) + O(x^30)) \\ Colin Barker, Apr 26 2019

G.f.: 1 -x*(2*x-1)*(x^2-4*x+1)/((4*x-1)*(x^3-6*x^2+5*x-1)). - Michael Somos, Jul 21 2002
a(n) = 9*a(n-1) - 26*a(n-2) + 25*a(n-3) - 4*a(n-4) for n>4. - Colin Barker, Apr 26 2019
14*a(n) = 4^n +2*(3*A005021(n) -10*A005021(n-1) +6*A005021(n-2)), n>0. - R. J. Mathar, Nov 05 2024

A215492 a(n) = 21a(n-2) + 7a(n-3), with a(0)=0, a(1)=3, and a(2)=6.

Original entry on oeis.org

0, 3, 6, 63, 147, 1365, 3528, 29694, 83643, 648270, 1964361, 14199171, 45789471, 311933118, 1060973088, 6871121775, 24463966674, 151720368891, 561841152579, 3357375513429, 12860706786396, 74437773850062, 293576471108319, 1653218198356074, 6686170310225133

Offset: 0

Roman Witula, Aug 13 2012

We have a(n)=B(n;3), where B(n;d), n=1,2,..., d \in C, denote one of the quasi-Fibonacci numbers defined in the comments to A121449 and in the Witula-Slota-Warzynski paper. Its conjugate sequences A(n;3) and C(n;3) are discussed in A121458 and A215484 respectively. Similarly as in A121458 we deduce that each of the following elements a(3*n), a(3*n+1), a(3*n+2) is divided by 3*7^n for every n=0,1,... .

G. C. Greubel, Table of n, a(n) for n = 0..1000
Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (0, 21, 7).

Cf. A121458, A215484, A121449, A085810, A215404, A077998, A006054, A033304, A052975, A094789, A005021, A121442, A121458.

I:=[0,3,6]; [n le 3 select I[n] else 21*Self(n-2)+7*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Sep 18 2015

LinearRecurrence[{0,21,7}, {0,3,6}, 50]
CoefficientList[Series[(3 x + 6 x^2)/(1 - 21 x^2 - 7 x^3), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 18 2015 *)

concat(0,Vec((3+6*x)/(1-21*x^2-7*x^3)+O(x^99))) \\ Charles R Greathouse IV, Oct 01 2012

a(n) = (1/7)*((c(1)-c(4))*(1+3*c(1))^n + (c(2)-c(1))*(1+3*c(2))^n + (c(4)-c(2))*(1+3*c(4))^n), where c(j):=2*cos(2*Pi*j/7) (for the proof see Witula-Slota-Warzynski paper).

G.f.: (3*x+6*x^2)/(1-21*x^2-7*x^3).

A259475 Array read by antidiagonals: row n gives coefficients of Taylor series expansion of 1/F_{n+1}(t), where F_i(t) is a Fibonacci polynomial defined by F_0=1, F_1=1, F_{i+1} = F_i-t*F_{i-1}.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 8, 8, 1, 0, 1, 5, 13, 21, 16, 1, 0, 1, 6, 19, 40, 55, 32, 1, 0, 1, 7, 26, 66, 121, 144, 64, 1, 0, 1, 8, 34, 100, 221, 364, 377, 128, 1, 0, 1, 9, 43, 143, 364, 728, 1093, 987, 256, 1, 0, 1, 10, 53, 196, 560, 1288, 2380, 3280, 2584, 512, 1, 0

Offset: 0

N. J. A. Sloane, Jul 03 2015

			The first few antidiagonals are:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  1,  0;
  1, 3,  4,  1,   0;
  1, 4,  8,  8,   1,   0;
  1, 5, 13, 21,  16,   1,  0;
  1, 6, 19, 40,  55,  32,  1, 0;
  1, 7, 26, 66, 121, 144, 64, 1, 0;
  ...
Square array starts:
  [0] 1, 0,  0,   0,    0,    0,     0,     0,      0,       0,       0, ...
  [1] 1, 1,  1,   1,    1,    1,     1,     1,      1,       1,       1, ...
  [2] 1, 2,  4,   8,   16,   32,    64,   128,    256,     512,    1024, ...
  [3] 1, 3,  8,  21,   55,  144,   377,   987,   2584,    6765,   17711, ...
  [4] 1, 4, 13,  40,  121,  364,  1093,  3280,   9841,   29524,   88573, ...
  [5] 1, 5, 19,  66,  221,  728,  2380,  7753,  25213,   81927,  266110, ...
  [6] 1, 6, 26, 100,  364, 1288,  4488, 15504,  53296,  182688,  625184, ...
  [7] 1, 7, 34, 143,  560, 2108,  7752, 28101, 100947,  360526, 1282735, ...
  [8] 1, 8, 43, 196,  820, 3264, 12597, 47652, 177859,  657800, 2417416, ...
  [9] 1, 9, 53, 260, 1156, 4845, 19551, 76912, 297275, 1134705, 4292145, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Noureddine Chair, Explicit Computations for the Intersection Numbers on Grassmannians, and on the Space of Holomorphic Maps from CP¹ into G_r(Cⁿ), arXiv:hep-th/9808170, 1998. (Cf. Table 4).
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.

The initial rows of the array are A000007, A000012, A000079, A001906, A003432, A005021, A094811, A094256.
A(n,n) gives A274969.
Cf. A309896.
A188843 is a variant without the first two rows and the first column, and the antidiagonals read in opposite direction.

F:= proc(n) option remember;
      `if`(n<2, 1, expand(F(n-1)-t*F(n-2)))
    end:
A:= (n, k)-> coeff(series(1/F(n+1), t, k+1), t, k):
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Jul 04 2015

F[n_] := F[n] = If[n<2, 1, Expand[F[n-1] - t*F[n-2]]]; A[n_, k_] := SeriesCoefficient[1/F[n+1], { t, 0, k}]; Table[A[d-k, k], {d, 0, 12}, {k, 0, d}] // Flatten (* Jean-François Alcover, Feb 17 2016, after Alois P. Heinz *)

@cached_function
def F(n, k):
    if k <  0: return 0
    if k == 0: return 1
    return sum((-1)^j*binomial(n-1-j,j+1)*F(n,k-2-2*j) for j in (0..(n-2)/2))
def A(n, k): return F(n+1, 2*k)
print([A(n-k, k) for n in (0..11) for k in (0..n)]) # Peter Luschny, Aug 21 2019

Let F(n, k) = Sum_{j=0..(n-2)/2} (-1)^j*binomial(n-1-j, j+1)*F(n, k-2-2*j) for k > 0; F(n, 0) = 1 and F(n, k) = 0 if k < 0. Then A(n, k) = F(n+1, 2*k). See [Shibukawa] and A309896. - Peter Luschny, Aug 21 2019

More terms from Alois P. Heinz, Jul 04 2015

A122589 Expansion of 1/(1 - 11x + 45x^2 - 84x^3 + 70x^4 - 21*x^5 + x^6).

Original entry on oeis.org

1, 11, 76, 425, 2109, 9709, 42504, 179630, 740025, 2991495, 11920740, 46981740, 183579396, 712493461, 2750450981, 10572046555, 40495806764, 154683305139, 589504177384, 2242448706435, 8517201473375, 32309383853565

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 19 2006

Previous name was: Sum_{n >= 0} a(n)*x^(2n) / 4^(n+6) = 1/(4096 - 11264*x^2 + 11520*x^4 - 5376*x^6 + 1120*x^8 - 84*x^10 + x^12).

Suggested by study of polynomials associated with the regular 13-gon.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-45,84,-70,21,-1).

Cf. A005021, A094256, A122588.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-11*x+45*x^2 -84*x^3+70*x^4-21*x^5+x^6) )); // G. C. Greubel, Nov 29 2021

A122589:= proc(n) coeftayl(1/(4096-11264*x^2+11520*x^4-5376*x^6+1120*x^8-84*x^10 +x^12), x=0,2*n); %*2^(2*n+12); end: seq(A122589(n), n=0..30); # R. J. Mathar, Sep 21 2007

m=12; p[x_]:= ExpandAll[x^m*ChebyshevU[m, 1/x]]; Table[ SeriesCoefficient[ Series[2^(n+m-1)*x/p[x], {x,0,30}], n], {n,1,30,2}]

def A122589_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-11*x+45*x^2-84*x^3+70*x^4-21*x^5+x^6) ).list()
A122589_list(30) # G. C. Greubel, Nov 29 2021

G.f.: 1/(1 - 11*x + 45*x^2 - 84*x^3 + 70*x^4 - 21*x^5 + x^6). - Colin Barker, Oct 16 2013

Edited by N. J. A. Sloane, Oct 02 2006
More terms from R. J. Mathar, Sep 21 2007
New name from Colin Barker, Oct 16 2013

A072844 Number of words of length 2n-1 generated by the two letters s and t that reduce to the identity 1 by using the relations sssssss=1, tt=1 and stst=1. The generators s and t along with the three relations generate the 14-element dihedral group D7.

Original entry on oeis.org

0, 0, 0, 1, 9, 55, 286, 1365, 6188, 27132, 116281, 490337, 2043275, 8439210, 34621041, 141290436, 574274008, 2326683921, 9402807817, 37923176863, 152705590518, 614111175965, 2467123420524, 9903167265124, 39725253489545

Offset: 1

Jamaine Paddyfoot and John W. Layman, Jul 24 2002

nonn

			The 9 words of length 9 are ssssssstt, sssssstts, sssssttss, ssssttsss, sssttssss, ssttsssss, sttssssss, ttsssssss, tssssssst. - _Sean A. Irvine_, Oct 31 2024

H.S.M. Coxeter and W.O.J. Moser, Generators and Relations for Discrete Groups, Fourth Edition, (p.134).

Haggai Liu, Enumerative Properties of Cogrowth Series on Free Products of Finite Groups, ACA 2021 Session on Algorithmic Combinatorics, 2021.

Cf. A072266.

Bisection of A377573.

a(n) = 9*a(n-1) - 26*a(n-2) + 25*a(n-3) - 4*a(n-4).
g.f.: x^4 / ((1 - 4*x)*(1 - 5*x + 6*x^2 - x^3)). - Colin Barker, Feb 24 2017
28*a(n) = 4^n -4*( 2*A005021(n) -9*A005021(n-1) +11*A005021(n-2) ). - R. J. Mathar, Nov 05 2024

cp's OEIS Frontend

A216054 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 1 or if k-n >= 6, T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A216235 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 2 or if k-n >= 5, T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A005022 Number of walks of length 2n+6 in the path graph P_7 from one end to the other.

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A005024 Number of walks of length 2n+8 in the path graph P_9 from one end to the other.

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A005025 Random walks.

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A072266 Number of words of length 2n generated by the two letters s and t that reduce to the identity 1 using the relations sssssss=1, tt=1 and stst=1. The generators s and t along with the three relations generate the 14-element dihedral group D7.

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A215492 a(n) = 21*a(n-2) + 7*a(n-3), with a(0)=0, a(1)=3, and a(2)=6.

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A215492 a(n) = 21a(n-2) + 7a(n-3), with a(0)=0, a(1)=3, and a(2)=6.

A122589 Expansion of 1/(1 - 11x + 45x^2 - 84x^3 + 70x^4 - 21*x^5 + x^6).