A002605 -id:A002605 - cp's OEIS frontend

A102591 a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*3^(n-k).

1, 6, 44, 328, 2448, 18272, 136384, 1017984, 7598336, 56714752, 423324672, 3159738368, 23584608256, 176037912576, 1313964867584, 9807567290368, 73204678852608, 546407161659392, 4078438577864704, 30441879976280064

Offset: 0

Paul Barry, Jan 22 2005

In general, Sum_{k=0..n} binomial(2n+1,2k)*r^(n-k) has g.f. (1-(r-1)x)/(1-2(r+1)+(r-1)^2x^2) and a(n) = ((sqrt(r)-1)^(2n+1) + (sqrt(r)+1)^(2n+1))/(2*sqrt(r)).

Index entries for linear recurrences with constant coefficients, signature (8,-4).

Bisection of A080879 or A002605.

Cf. A066443, A079935, A099156, A102592, A107903.

LinearRecurrence[{8,-4},{1,6},20] (* Harvey P. Dale, Sep 28 2021 *)

G.f.: (1-2x)/(1-8x+4x^2);
a(n) = 8*a(n-1) - 4*a(n-2);
a(n) = sqrt(3)*(sqrt(3)-1)^(2n+1)/6 + sqrt(3)*(sqrt(3)+1)^(2n+1)/6.
a(n) = 2^n*A079935(n). - R. J. Mathar, Sep 20 2012
a(n) = 2^(2*n+1)*Sum_{k >= n} binomial(2*k,2*n)*(1/3)^(k+1). Cf. A099156. - Peter Bala, Nov 29 2021
3*a(n)^2 = A107903(n)^2 + 2^(2*n+1). - Philippe Deléham, Mar 21 2023

A108898 a(n+3) = 3a(n+2) - 2a(n), a(0) = -1, a(1) = 1, a(2) = 3.

Original entry on oeis.org

-1, 1, 3, 11, 31, 87, 239, 655, 1791, 4895, 13375, 36543, 99839, 272767, 745215, 2035967, 5562367, 15196671, 41518079, 113429503, 309895167, 846649343, 2313089023, 6319476735, 17265131519, 47169216511, 128868696063, 352075825151, 961889042431, 2627929735167, 7179637555199

Offset: 0

Creighton Dement, Jul 16 2005

In reference to the program code, "ibasek" corresponds to the floretion 'ik'. Sequences in this same batch are "kbase" = A005665 (Tower of Hanoi with cyclic moves only.) and "ibase" = A077846.

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

Cf. A005665, A077846, A028860.

Cf. A002605, A026150, A028859, A030195, A080040, A083337, A106435, A125145.

a108898 n = a108898_list !! n
a108898_list = -1 : 1 : 3 :
   zipWith (-) (map (* 3) $ drop 2 a108898_list) (map (* 2) a108898_list)
-- Reinhard Zumkeller, Oct 15 2011

seriestolist(series((-1+4*x)/((x-1)*(2*x^2+2*x-1)), x=0,31)); -or- Floretion Algebra Multiplication Program, FAMP Code: 2ibaseksumseq[A*B] with A = + 'i + 'ii' + 'ij' + 'ik' and B = + .5'i + .5'j - .5'k + .5i' - .5j' + .5k' + .5'ij' + .5'ik' - .5'ji' - .5'ki'; Sumtype is set to:sum[(Y[0], Y[1], Y[2]),mod(3)

LinearRecurrence[{3, 0, -2}, {-1, 1, 3}, 40] (* Paolo Xausa, Aug 21 2024 *)

Vec(-(1 - 4*x) / ((1 - x)*(1 - 2*x - 2*x^2)) + O(x^40)) \\ Colin Barker, Apr 29 2019

a(n) = A028860(n+2)-1.
G.f.: (-1+4*x)/((x-1)*(2*x^2+2*x-1)).
From Colin Barker, Apr 29 2019: (Start)
a(n) = (-1 + (-(1-sqrt(3))^n + (1+sqrt(3))^n)/sqrt(3)).
a(n) = 3*a(n-1) - 2*a(n-3) for n>2.
(End)

A202064 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 4, 0, 0, 5, 10, 1, 0, 0, 6, 20, 6, 0, 0, 0, 7, 35, 21, 1, 0, 0, 0, 8, 56, 56, 8, 0, 0, 0, 0, 9, 84, 126, 36, 1, 0, 0, 0, 0, 10, 120, 252, 120, 10, 0, 0, 0, 0, 0, 11, 165, 462, 330, 55, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 10 2011

Riordan array (x/(1-x)^2, x^2/(1-x)^2).
Mirror image of triangle in A119900.
A203322*A130595 as infinite lower triangular matrices. - Philippe Deléham, Jan 05 2011
From Gus Wiseman, Jul 07 2025: (Start)
Also the number of subsets of {1..n} containing n with k maximal runs (sequences of consecutive elements increasing by 1). For example, row n = 5 counts the following subsets:
  {5}          {1,5}      {1,3,5}
  {4,5}        {2,5}
  {3,4,5}      {3,5}
  {2,3,4,5}    {1,2,5}
  {1,2,3,4,5}  {1,4,5}
               {2,3,5}
               {2,4,5}
               {1,2,3,5}
               {1,2,4,5}
               {1,3,4,5}
For anti-runs instead of runs we have A053538.
Without requiring n see A210039, A202023, reverse A098158, A109446.
(End)

			Triangle begins :
1
2, 0
3, 1, 0
4, 4, 0, 0
5, 10, 1, 0, 0
6, 20, 6, 0, 0, 0
7, 35, 21, 1, 0, 0, 0
8, 56, 56, 8, 0, 0, 0, 0

Cf. A007318, A005314 (antidiagonal sums), A119900, A084938, A130595, A203322.
Column k = 1 is A000027.
Row sums are A000079.
Column k = 2 is A000292.
Without zeros we have A034867.
Last nonzero term in each row appears to be A124625.
A034839 counts subsets by number of maximal runs, for anti-runs A384893.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
Cf. A000045, A000071, A001629, A010027, A053538, A208342, A210034, A245563, A268193, A384177, A384890.
Cf. A098158, A109446, A202023, A210039.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Length[Split[#,#2==#1+1&]]==k&]],{n,12},{k,n}] (* Gus Wiseman, Jul 07 2025 *)

G.f.: 1/((1-x)^2-y*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000027(n+1), A000079(n), A000129(n+1), A002605(n+1), A015518(n+1), A063727(n), A002532(n+1), A083099(n+1), A015519(n+1), A003683(n+1), A002534(n+1), A083102(n), A015520(n+1), A091914(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 11, 12, 13 respectively.
T(n,k) = binomial(n+1,2k+1).
T(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 15 2012

A231131 T(n,k) = Number of (n+1) X (k+1) white-square subarrays of 0..2 arrays with no element equal to a strict majority of its diagonal and antidiagonal neighbors, with values 0..2 introduced in row major order.

Original entry on oeis.org

1, 2, 2, 6, 8, 6, 16, 40, 40, 16, 44, 176, 308, 176, 44, 120, 808, 2260, 2260, 808, 120, 328, 3584, 16812, 27664, 16812, 3584, 328, 896, 16368, 124644, 336004, 336004, 124644, 16368, 896, 2448, 72640, 924900, 4150352, 6794904, 4150352, 924900, 72640, 2448

Offset: 1

R. H. Hardin, Nov 04 2013

Table starts
..1...2.....6.....16......44.......120........328.........896..........2448
..2...8....40....176.....808......3584......16368.......72640........331648
..6..40...308...2260...16812....124644.....924900.....6862052......50913012
.16.176..2260..27664..336004...4150352...50257244...621150768....7520563372
.44.808.16812.336004.6794904.137063228.2766762720.55844298404.1127200291672

			Some solutions for n=2, k=4
..0..x..1..x..1....0..x..0..x..1....0..x..1..x..0....0..x..0..x..1
..x..1..x..2..x....x..1..x..0..x....x..1..x..2..x....x..1..x..2..x
..2..x..2..x..1....2..x..1..x..1....0..x..0..x..0....0..x..2..x..1

R. H. Hardin, Table of n, a(n) for n = 1..312

Column 1 is A002605.

Empirical for column k:
k=1: a(n) = 2*a(n-1) +2*a(n-2);
k=2: a(n) = 22*a(n-2) -36*a(n-4) +16*a(n-6);
k=3: [order 8];
k=4: [order 18, even terms];
k=5: [order 34];
k=6: [order 90, even terms].

A265278 Expansion of (x^4+x^3-x^2+x)/(x^3+x^2-3*x+1).

Original entry on oeis.org

0, 1, 2, 6, 16, 40, 98, 238, 576, 1392, 3362, 8118, 19600, 47320, 114242, 275806, 665856, 1607520, 3880898, 9369318, 22619536, 54608392, 131836322, 318281038, 768398400, 1855077840, 4478554082, 10812186006, 26102926096, 63018038200, 152139002498, 367296043198

Offset: 0

Alois P. Heinz, Apr 06 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Marika Diepenbroek, Monica Maus, and Alex Stoll, Pattern Avoidance in Reverse Double Lists, Preprint 2015. See Table 3.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 41.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Agrees with A213667 except for initial terms.

Cf. A002605, A265106, A265107, A270810.

Table[2 Fibonacci[n-1, 2] + LucasL[n-1, 2]/2 + KroneckerDelta[n-1] - 1, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)
LinearRecurrence[{3,-1,-1},{0,1,2,6,16},40] (* Harvey P. Dale, Mar 18 2018 *)

concat(0, Vec(x*(1-x+x^2+x^3)/((1-x)*(1-2*x-x^2)) + O(x^50))) \\ Colin Barker, Apr 12 2016

From Colin Barker, Apr 12 2016: (Start)
a(n) = (-2 + (1-sqrt(2))^n + (1+sqrt(2))^n)/2 for n>1.
a(n) = 3*a(n-1)-a(n-2)-a(n-3) for n>4.
(End)
E.g.f.: x + (cosh(sqrt(2)*x) - 1)*exp(x). - Ilya Gutkovskiy, Sep 16 2016

A096608 Triangle read by rows: T(n,k)=number of Catalan knight paths in right half-plane from (0,0) to (n,k), for 0 <= k <= 2n, n >= 0. (See A096587 for the definition of a Catalan knight.)

Original entry on oeis.org

1, 0, 0, 1, 2, 1, 0, 0, 1, 0, 2, 3, 2, 0, 0, 1, 8, 6, 1, 3, 4, 3, 0, 0, 1, 6, 12, 16, 12, 3, 4, 5, 4, 0, 0, 1, 44, 33, 18, 21, 27, 20, 6, 5, 6, 5, 0, 0, 1, 60, 76, 95, 72, 40, 34, 41, 30, 10, 6, 7, 6, 0, 0, 1, 256, 210, 154, 155, 177, 135, 75, 52, 58, 42, 15, 7, 8, 7, 0, 0, 1, 460, 520, 581, 480

Offset: 0

Clark Kimberling, Jun 29 2004

			Rows:
  1;
  0, 0, 1;
  2, 1, 0, 0, 1;
  0, 2, 3, 2, 0, 0, 1;
T(3,2) counts these paths:
  (0,0)-(1,-2)-(2,0)-(3,2);
  (0,0)-(1,2)-(2,0)-(3,2);
  (0,0)-(1,2)-(2,4)-(3,2).

Paolo Xausa, Table of n, a(n) for n = 0..9999 (rows 0..99 of triangle, flattened)
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024.
Rémy Sigrist, Representation of the odd terms for n = 0..2^11

Cf. A096587, A096609, A096610, A096611, A096612, A355320.

A096608[rowmax_]:=Module[{T},T[0,0]=1;T[n_,k_]:=T[n,k]=If[k<=2n,T[n-1,Abs[k-2]]+T[n-2,Abs[k-1]]+T[n-1,k+2]+T[n-2,k+1],0];Table[T[n,k],{n,0,rowmax},{k,0,2n}]]; A096608[10] (* Generates 11 rows *) (* Paolo Xausa, May 09 2023 *)

row(n) = { my (rr=0, r=1); for (k=1, n, [rr, r]=[r, r*(1+'X^4)+rr*('X^3+'X^5)]); Vec(r)[1+2*n..1+4*n] } \\ Rémy Sigrist, Jun 29 2022

T(0, 0) = 1, T(0, 1) = 0, T(0, 2) = 0; T(1, 0) = 0, T(1, 1) = 0, T(1, 2) = 1.
For n >= 2, T(n, 0) = 2*T(n-2, 1) + 2*T(n-1, 2); T(n, 1) = T(n-2, 0) + T(n-2, 2) + T(n-1, 3) + T(n-1, 1); for 2 <= k <= 2n, T(n, k) = T(n-2, k-1) + T(n-2, k+1) + T(n-1, k-2) + T(n-1, k+2).
T(n, 0) + 2*Sum_{k = 1..2*n} T(n, k) = A002605(k). - Rémy Sigrist, Jun 29 2022

Offset changed to 0 by Rémy Sigrist, Jun 29 2022

A129267 Triangle with T(n,k) = T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k) and T(0,0)=1 .

Original entry on oeis.org

1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -1, -3, -2, 1, 1, 0, -2, -5, -3, 1, 1, 1, 2, -2, -7, -4, 1, 1, 1, 5, 7, -1, -9, -5, 1, 1, 0, 3, 12, 15, 1, -11, -6, 1, 1, -1, -3, 3, 21, 26, 4, -13, -7, 1, 1, -1, -7, -15, -3, 31, 40, 8, -15, -8, 1, 1

Offset: 0

Philippe Deléham, Jun 08 2007

Triangle T(n,k), 0<=k<=n, read by rows given by [1,-1,1,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] where DELTA is the operator defined in A084938 . Riordan array (1/(1-x+x^2),(x*(1-x))/(1-x+x^2)); inverse array is (1/(1+x),(x/(1+x))*c(x/(1+x))) where c(x)is g.f. of A000108 .

Row sums are ( with the addition of a first row {0}): 0, 1, 2, 2, 0, -4, -8, -8, 0, 16, 32,... (see A009545). - Roger L. Bagula, Nov 15 2009

			Triangle begins:
   1;
   1,  1;
   0,  1,   1;
  -1, -1,   1,  1;
  -1, -3,  -2,  1,  1;
   0, -2,  -5, -3,  1,   1;
   1,  2,  -2, -7, -4,   1,   1;
   1,  5,   7, -1, -9,  -5,   1,   1;
   0,  3,  12, 15,  1, -11,  -6,   1,  1;
  -1, -3,   3, 21, 26,   4, -13,  -7,  1, 1;
  -1, -7, -15, -3, 31,  40,   8, -15, -8, 1, 1;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A063967, A167925, A199324, A202503, A202551.

T:= proc(n, k) option remember;
      if k<0 or  k>n  then 0
    elif n=0 and k=0 then 1
    else T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 14 2020

m = {{a, 1}, {-1, 1}}; v[0]:= {0, 1}; v[n_]:= v[n] = m.v[n-1]; Table[CoefficientList[v[n][[1]], a], {n, 0, 10}]//Flatten (* Roger L. Bagula, Nov 15 2009 *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n==0 && k==0, 1, T[n-1, k-1] + T[n-1, k] - T[n-2, k-1] - T[n-2, k] ]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 14 2020 *)

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (n==0 and k==0): return 1
    else: return T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 14 2020

Sum{k=0..n} T(n,k)*x^k = { (-1)^n*A057093(n), (-1)^n*A057092(n), (-1)^n*A057091(n), (-1)^n*A057090(n), (-1)^n*A057089(n), (-1)^n*A057088(n), (-1)^n*A057087(n), (-1)^n*A030195(n+1), (-1)^n*A002605(n), A039834(n+1), A000007(n), A010892(n), A099087(n), A057083(n), A001787(n+1), A030191(n), A030192(n), A030240(n), A057084(n), A057085(n), A057086(n) } for x=-11, -10, ..., 8, 9, respectively .
Sum{k=0..n} T(n,k)*A000045(k) = A100334(n).
Sum{k=0..floor(n/2)} T(n-k,k) = A050935(n+2).
T(n,k)= Sum{j>=0} A109466(n,j)*binomial(j,k).
T(n,k) = (-1)^(n-k)*A199324(n,k) = (-1)^k*A202551(n,k) = A202503(n,n-k). - Philippe Deléham, Mar 26 2013
G.f.: 1/(1-x*y+x^2*y-x+x^2). - R. J. Mathar, Aug 11 2015

Riordan array definition corrected by Ralf Stephan, Jan 02 2014

A221459 T(n,k)=Number of 0..k arrays of length n with each element unequal to at least one neighbor, with new values introduced in 0..k order.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 2, 6, 3, 0, 1, 2, 7, 16, 5, 0, 1, 2, 7, 24, 44, 8, 0, 1, 2, 7, 25, 88, 120, 13, 0, 1, 2, 7, 25, 101, 328, 328, 21, 0, 1, 2, 7, 25, 102, 436, 1235, 896, 34, 0, 1, 2, 7, 25, 102, 455, 1971, 4668, 2448, 55, 0, 1, 2, 7, 25, 102, 456, 2192, 9159, 17675

Offset: 1

R. H. Hardin Jan 17 2013

Table starts
..0.....0......0......0.......0.......0.......0.......0.......0.......0.......0
..1.....1......1......1.......1.......1.......1.......1.......1.......1.......1
..1.....2......2......2.......2.......2.......2.......2.......2.......2.......2
..2.....6......7......7.......7.......7.......7.......7.......7.......7.......7
..3....16.....24.....25......25......25......25......25......25......25......25
..5....44.....88....101.....102.....102.....102.....102.....102.....102.....102
..8...120....328....436.....455.....456.....456.....456.....456.....456.....456
.13...328...1235...1971....2192....2218....2219....2219....2219....2219....2219
.21...896...4668...9159...11203...11605...11639...11640...11640...11640...11640
.34..2448..17675..43262...59814...64647...65320...65363...65364...65364...65364
.55..6688..66974.206285..329343..379349..389533..390592..390645..390646..390646
.89.18272.253858.988963.1851911.2320555.2451393.2471066.2472654.2472718.2472719

			Some solutions for n=6 k=4
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..1....1....1....1....1....1....1....1....1....1....1....1....1....1....1....1
..2....2....2....2....1....0....0....2....2....2....1....1....2....2....2....1
..3....3....1....2....2....2....2....3....2....1....2....2....2....0....1....0
..4....3....0....1....2....0....3....1....3....3....0....3....0....2....2....1
..1....4....3....3....3....3....2....0....1....0....1....2....3....1....1....0

R. H. Hardin, Table of n, a(n) for n = 1..2080

Column 1 is A000045(n-1)

Column 2 is A002605(n-1)

A074358 Coefficient of q^1 in nu(n), where nu(0)=1, nu(1)=b and, for n >= 2, nu(n) = bnu(n-1) + lambda(1 + q + q^2 + ... + q^(n-2))*nu(n-2) with (b,lambda)=(2,2).

Original entry on oeis.org

0, 0, 0, 4, 20, 80, 288, 976, 3184, 10112, 31488, 96576, 292672, 878336, 2614784, 7731456, 22728448, 66482176, 193617920, 561718272, 1624101888, 4681535488, 13457924096, 38592008192, 110419341312, 315287830528, 898583560192

Offset: 0

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

nonn

Coefficient of q^0 is A002605.

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=6, nu(3) = 16 + 4q, nu(4) = 44 + 20q + 12q^2, nu(5) = 120 + 80q + 64q^2 + 40q^3 + 8q^4, so the coefficients of q^1 are 0,0,0,4,20,80.

M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.

Coefficient of q^0, q^2 and q^3 are in A002605, A074359 and A074360. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074357, A074361-A074363.

nu := proc(b,lambda,n) global q; local qp,i ; if n = 0 then RETURN(1) ; elif n =1 then RETURN(b) ; fi ; qp:=0 ; for i from 0 to n-2 do qp := qp + q^i ; od ; RETURN( b*nu(b,lambda,n-1)+lambda*qp*nu(b,lambda,n-2)) ; end: A074358 := proc(n) RETURN( coeftayl(nu(2,2,n),q=0,1) ) ; end: for n from 0 to 30 do printf("%d,", A074358(n)) ; od ; # R. J. Mathar, Sep 20 2006

nu[0] = 1; nu[1] = 2; nu[n_] := nu[n] = 2*nu[n-1] + 2*Total[q^Range[0, n-2] ]*nu[n-2] // Expand;
a[n_] := Coefficient[nu[n], q, 1];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Nov 17 2017 *)

G.f.: 4*x^3*(x + 1)/(2*x^2 + 2*x - 1)^2 (conjectured). - Chai Wah Wu, May 30 2016
a(n) = (1/18)*((1 + sqrt(3))^n*(-9 + 2*sqrt(3)) - (1 - sqrt(3))^n*(9 + 2*sqrt(3)) + 3*((1 - sqrt(3))^n + (1 + sqrt(3))^n)*n) for n > 0 (conjectured). - Colin Barker, Nov 17 2017
a(n) = 4*a(n-1) - 8*a(n-3) - 4*a(n-4) for n > 4 (conjectured). - Colin Barker, Nov 17 2017

More terms from R. J. Mathar, Sep 20 2006

A074360 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=bnu(n-1)+lambda(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(2,2).

Original entry on oeis.org

0, 0, 0, 0, 0, 40, 232, 1072, 4400, 16864, 61728, 218496, 753792, 2547840, 8468608, 27755776, 89886976, 288101888, 915089920, 2883416064, 9021001728, 28042881024, 86672025600, 266472878080, 815347462144, 2483820617728

Offset: 0

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

nonn

Coefficient of q^0 is A002605.

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=6, nu(3)=16+4q, nu(4)=44+20q+12q^2, nu(5)=120+80q+64q^2+40q^3+8q^4, so the coefficients of q^1 are 0,0,0,0,0,40.

M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.

Coefficient of q^0, q^1 and q^2 are in A002605, A074358 and A074359. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074357, A074361-A074363.

nu := proc(b,lambda,n) global q; local qp,i ; if n = 0 then RETURN(1) ; elif n =1 then RETURN(b) ; fi ; qp:=0 ; for i from 0 to n-2 do qp := qp + q^i ; od ; RETURN( b*nu(b,lambda,n-1)+lambda*qp*nu(b,lambda,n-2)) ; end: A074360 := proc(n) RETURN( coeftayl(nu(2,2,n),q=0,3) ) ; end: for n from 0 to 30 do printf("%d,", A074360(n)) ; od ; # R. J. Mathar, Sep 20 2006

nu[0] = 1; nu[1] = 2; nu[n_] := nu[n] = 2*nu[n - 1] + 2*Total[q^Range[0, n - 2]]*nu[n - 2] // Expand;
a[n_] := Coefficient[nu[n], q, 3];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 18 2017 *)

Conjecture: O.g.f: 8*x^5*(1+x)*(12*x^4+24*x^3-2*x^2-16*x+5)/(2*x^2+2*x-1)^4. - R. J. Mathar, Jul 22 2009

More terms from R. J. Mathar, Sep 20 2006

cp's OEIS Frontend

A102591 a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*3^(n-k).

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A108898 a(n+3) = 3*a(n+2) - 2*a(n), a(0) = -1, a(1) = 1, a(2) = 3.

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A202064 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A231131 T(n,k) = Number of (n+1) X (k+1) white-square subarrays of 0..2 arrays with no element equal to a strict majority of its diagonal and antidiagonal neighbors, with values 0..2 introduced in row major order.

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A265278 Expansion of (x^4+x^3-x^2+x)/(x^3+x^2-3*x+1).

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A096608 Triangle read by rows: T(n,k)=number of Catalan knight paths in right half-plane from (0,0) to (n,k), for 0 <= k <= 2n, n >= 0. (See A096587 for the definition of a Catalan knight.)

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A129267 Triangle with T(n,k) = T(n-1,k-1) + T(n-1,k) - T(n-2,k-1) - T(n-2,k) and T(0,0)=1 .

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A221459 T(n,k)=Number of 0..k arrays of length n with each element unequal to at least one neighbor, with new values introduced in 0..k order.

A108898 a(n+3) = 3a(n+2) - 2a(n), a(0) = -1, a(1) = 1, a(2) = 3.

A074358 Coefficient of q^1 in nu(n), where nu(0)=1, nu(1)=b and, for n >= 2, nu(n) = bnu(n-1) + lambda(1 + q + q^2 + ... + q^(n-2))*nu(n-2) with (b,lambda)=(2,2).

A074360 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=bnu(n-1)+lambda(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(2,2).