A122367 -id:A122367 - cp's OEIS frontend

A208343 Triangle of coefficients of polynomials v(n,x) jointly generated with A208342; see the Formula section.

1, 0, 2, 0, 1, 3, 0, 1, 2, 5, 0, 1, 2, 5, 8, 0, 1, 2, 6, 10, 13, 0, 1, 2, 7, 13, 20, 21, 0, 1, 2, 8, 16, 29, 38, 34, 0, 1, 2, 9, 19, 39, 60, 71, 55, 0, 1, 2, 10, 22, 50, 86, 122, 130, 89, 0, 1, 2, 11, 25, 62, 116, 187, 241, 235, 144, 0, 1, 2, 12, 28, 75, 150, 267, 392, 468

Offset: 1

Clark Kimberling, Feb 25 2012

u(n,n) = A000045(n+1) (Fibonacci numbers).
n-th row sum:  2^(n-1)
As triangle T(n,k) with 0 <= k <= n, it is (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012

			First five rows:
  1;
  0, 2;
  0, 1, 3;
  0, 1, 2, 5;
  0, 1, 2, 5, 8;
First five polynomials v(n,x):
  1
     2x
      x + 3x^2
      x + 2x^2 + 5x^3
      x + 2x^2 + 5x^3 + 8x^4.

Cf. A208343.

Cf. A084938, A000045, A000079.

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] + 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[%]  (* A208342 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208343 *)

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = x*u(n-1,x) + x*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) with 0 <= k <= n:
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k > n or if k < 0.
G.f.: (1-(1-y)*x)/(1-(1+y)*x+y*(1-y)*x^2).
Sum_{k=0..n} T(n,k)*x^k = (-1)*A091003(n+1), A152166(n), A000007(n), A000079(n), A055099(n), A152224(n) for x = -2, -1, 0, 1, 2, 3 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A087205(n), A140165(n+1), A016116(n+1), A000045(n+2), A000079(n), A122367(n), A006012(n), A052961(n), A154626(n) for x = -3, -2, -1, 0, 1, 2, 3, 4 respectively. (End)
T(n,k) = A208748(n,k)/2^k. - Philippe Deléham, Mar 05 2012

A122392 Dimension of 3-variable non-commutative harmonics (Hausdorff derivative). The dimension of the space of non-commutative polynomials in 3 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( w ) = sum over all subwords of w deleting xi once).

Original entry on oeis.org

1, 2, 5, 15, 46, 139, 416, 1248, 3744, 11232, 33696, 101088, 303264, 909792, 2729376, 8188128, 24564384, 73693152, 221079456, 663238368, 1989715104, 5969145312, 17907435936, 53722307808, 161166923424, 483500770272, 1450502310816

Offset: 0

Mike Zabrocki, Aug 31 2006

			a(1) = 2 because x1 - x2, x2 - x3 are killed by d_x1 + d_x2 + d_x3
a(2) = 5 because x1 x2 - x2 x1, x1 x3 - x3 x1, x2 x3 - x3 x2, 2 x1 x2 - x2 x2 - 2 x1 x3 + x3 x3,
x1 x1 - 2 x2 x1 + 2 x2 x3 - x3 x3 are killed by d_x1 + d_x2 + d_x3, d_x1^2 + d_x2^2 + d_x3^2 and
d_x1 d_x2 + d_x1 d_x3 + d_x2 d_x3

C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.
C. Reutenauer, Free Lie algebras. London Mathematical Society Monographs. New Series, 7. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. xviii+269 pp.

N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/0502082 , Canad. J. Math. 60 (2008), no. 2, 266-296.

Cf. A118264, A122367, A122391, A122393, A122394.

coeffs(convert(series(mul(1-q^i,i=1..3)/(1-3*q),q,20),`+`)-O(q^20),q);

G.f.: (1-q)*(1-q^2)*(1-q^3)/(1-3*q) 3^n - 3^(n-1) - 3^(n-2) + 3^(n-4) + 3^(n-5) - 3^(n-6) (for n>5) a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 15, a(4) = 46, a(5) = 139, a(n) = 416*3^(n-6) for n>5

A373639 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-5k,k).

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 9, 16, 27, 43, 71, 119, 196, 322, 533, 882, 1455, 2402, 3970, 6558, 10829, 17887, 29548, 48804, 80608, 133146, 219925, 363254, 599998, 991044, 1636944, 2703794, 4465957, 7376591, 12184181, 20125051, 33241289, 54905857, 90690002, 149795989

Offset: 0

Seiichi Manyama, Jun 12 2024

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,2,0,0,-1).

Cf. A000930, A373640.

Cf. A052535, A122367, A373644.

LinearRecurrence[{1,0,2,0,0,-1},{1, 1, 1, 2, 4, 6},40] (* James C. McMahon, Jun 14 2024 *)
Table[Sum[Binomial[2n-5k,k],{k,0,Floor[n/3]}],{n,0,40}] (* Harvey P. Dale, Sep 29 2024 *)

a(n) = sum(k=0, n\3, binomial(2*n-5*k, k));

G.f.: 1 / (1 - x^3 - x/(1 - x^3)).

a(n) = a(n-1) + 2*a(n-3) - a(n-6) for n > 5.

A155110 a(n) = 8*Fibonacci(2n+1).

Original entry on oeis.org

8, 16, 40, 104, 272, 712, 1864, 4880, 12776, 33448, 87568, 229256, 600200, 1571344, 4113832, 10770152, 28196624, 73819720, 193262536, 505967888, 1324641128, 3467955496, 9079225360, 23769720584, 62229936392, 162920088592, 426530329384, 1116670899560

Offset: 0

Paul Curtz, Jan 20 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A001519, A027941, A099496, A122367, A153873, A154811, A156551.

[8*Fibonacci(2*n+1): n in [0..30]]; // Vincenzo Librandi, Aug 07 2011

A155110 := proc(n) 8*combinat[fibonacci](2*n+1) ; end: seq(A155110(n),n=0..50) ; # R. J. Mathar, Oct 06 2009

8*Fibonacci[2*Range[0,30]+1] (* G. C. Greubel, Apr 21 2021 *)

[8*fibonacci(2*n+1) for n in (0..30)] # G. C. Greubel, Apr 21 2021

a(n) = 8*A001519(n+1) = 8*A122367(n) = 8 *|A099496(n)|.
a(n) == A154811(n+6) (mod 9).
a(n) == A156551(n) (mod 10).
a(n) = A153873(n) - A027941(n).
G.f.: 8*(1 - x)/(1 - 3*x + x^2). - G. C. Greubel, Apr 21 2021

Comments converted to formulas by R. J. Mathar, Oct 06 2009

A210215 Triangle of coefficients of polynomials u(n,x) jointly generated with A210216; see the Formula section.

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 2, 5, 7, 1, 2, 5, 12, 11, 1, 2, 5, 13, 26, 16, 1, 2, 5, 13, 33, 51, 22, 1, 2, 5, 13, 34, 79, 92, 29, 1, 2, 5, 13, 34, 88, 176, 155, 37, 1, 2, 5, 13, 34, 89, 221, 365, 247, 46, 1, 2, 5, 13, 34, 89, 232, 530, 709, 376, 56, 1, 2, 5, 13, 34, 89, 233, 596

Offset: 1

Clark Kimberling, Mar 19 2012

Limiting row:  odd-indexed Fibonacci numbers, (A122367, A001519)
n-th row sum: -1+2^n
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
2...1
2...4...1
2...5...7....1
2...5...12...11...1
First three polynomials u(n,x): 1, 2 + x, 2 + 4x + x^2.

Cf. A210216, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + 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[%]    (* A210215 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210216 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)

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

A052568 E.g.f.: (1-x)/(1-3*x+x^2).

Original entry on oeis.org

1, 2, 10, 78, 816, 10680, 167760, 3074400, 64391040, 1517201280, 39720844800, 1143895737600, 35937095040000, 1223098971494400, 44829605505484800, 1760481463732992000, 73744004937867264000, 3282093293695856640000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Laguerre transform of n!Fibonacci(n+1), A005442. - Paul Barry, Aug 08 2008

G. C. Greubel, Table of n, a(n) for n = 0..385
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 510

Apart from signs, row sums of A079461.

Cf. A005442, A122367.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1-x)/(1-3*x+x^2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 23 2018

spec := [S,{S=Sequence(Union(Z,Prod(Z,Sequence(Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
a:= n-> n! * (Matrix([[1,1]]). Matrix([[3,1], [ -1,0]])^n)[1,1]: seq(a(n), n=0..20); # Alois P. Heinz, Jun 01 2009

With[{nn=20},CoefficientList[Series[(1-x)/(1-3x+x^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 07 2012 *)
Table[Fibonacci[2n+1] n!, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 29 2015 *)

x='x+O('x^30); Vec(serlaplace((1-x)/(1-3*x+x^2))) \\ G. C. Greubel, May 23 2018

Recurrence: {a(0)=1, a(1)=2, (n^2+3*n+2)*a(n)+(-6-3*n)*a(n+1)+a(n+2)=0.}
a(n) = Sum(1/5*(1+_alpha)*_alpha^(-1-n), _alpha=RootOf(_Z^2-3*_Z+1))*n!.
a(n) = Sum_{k=0..n} binomial(n,k)(n!/k!)*k!Fibonacci(k+1). - Paul Barry, Aug 08 2008
a(n) = n!*A122367(n). - R. J. Mathar, Nov 27 2011

Edited by N. J. A. Sloane, May 29 2009

A104726 Triangle generated as the matrix product of A026729 and A000012 (triangular views), read by rows.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 3, 3, 1, 5, 5, 5, 4, 1, 8, 8, 8, 8, 5, 1, 13, 13, 13, 13, 12, 6, 1, 21, 21, 21, 21, 21, 17, 7, 1, 34, 34, 34, 34, 34, 33, 23, 8, 1, 55, 55, 55, 55, 55, 55, 50, 30, 9, 1, 89, 89, 89, 89, 89, 89, 88, 73, 38

Offset: 0

Gary W. Adamson, Mar 20 2005

If the triangular factors A026729 and A000012 are commuted in the product, A004070 results.

Riordan array (1/(1-x-x^2), x*(1+x)). - Philippe Deléham, Mar 06 2013

			First few rows of the triangle are
1;
1, 1;
2, 2, 1;
3, 3, 3, 1;
5, 5, 5, 4, 1;
8, 8, 8, 8, 5, 1;
13, 13, 13, 13, 12, 6, 1;
21, 21, 21, 21, 21, 17, 7, 1;
...
Production array begins
1, 1
1, 1, 1
-1, -1, 1, 1
2, 2, -1, 1, 1
-5, -5, 2, -1, 1, 1
14, 14, -5, 2, -1, 1, 1
-42, -42, 14, -5, 2, -1, 1, 1
132, 132, -42, 14, -5, 2, -1, 1, 1
-429, -429, 132, -42, 14, -5, 2, -1, 1, 1
... which is based on A000108 or A168491. - _Philippe Deléham_, Mar 06 2013

Cf. A001629 (row sums), A026729, A004070, A000071.

A104726 := proc(n,k)
        add( binomial(j,n-j),j=k..n) ;
end proc:
seq(seq(A104726(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Oct 30 2011

T(n,k) = sum_{j=k..n} binomial(j,n-j). - R. J. Mathar, Oct 30 2011
T(n,0) = T(n-1,0) + T(n-2,0), T(n,k) = T(n-1,k-1) + T(n-2,k-1) for k>0. - Philippe Deléham, Mar 06 2013
T(2*n,n) = A000045(2n+1) = A001519(n+1) = A122367(n). - Philippe Deléham, Mar 06 2013

A164267 A Fibonacci convolution.

Original entry on oeis.org

0, 1, 2, 7, 16, 46, 114, 309, 792, 2101, 5456, 14356, 37468, 98281, 256998, 673323, 1761984, 4614226, 12078110, 31624285, 82787980, 216750601, 567446112, 1485616392, 3889356696, 10182528721, 26658108074, 69791991919, 182717549872

Offset: 0

Paul Barry, Aug 11 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-4,1)

I:=[0,1,2,7]; [n le 4 select I[n] else 2*Self(n-1)+3*Self(n-2)-4*Self(n-3)+Self(n-4): n in [1..30]]; // Vincenzo Librandi, Sep 13 2017

LinearRecurrence[{2,3,-4,1},{0,1,2,7},30] (* Harvey P. Dale, Jul 12 2011 *)
CoefficientList[Series[x / ((1 + x - x^2) (1 - 3 x + x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 13 2017 *)

x='x+O('x^50); concat([0], Vec(x/((1+x-x^2)*(1-3*x+x^2)))) \\ G. C. Greubel, Sep 12 2017

G.f.: x/((1+x-x^2)(1-3x+x^2)).
a(n) = Sum_{k=0..n} (-1)^k*F(k+1)*F(2(n-k)).
a(n) = Sum_{k=0..n} C(n,k)*F(k+1)*(1-(-1)^(n-k))/2.
a(n) = 2*a(n-1) + 3*a(n-2) - 4*a(n-3) + a(n-4).
a(n) = (A122367(n) - A039834(n-1))/2. - R. J. Mathar, Aug 17 2009

A202396 Triangle T(n,k), read by rows, given by (2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -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, 2, 5, 8, 3, 13, 27, 19, 5, 34, 86, 86, 42, 8, 89, 265, 338, 234, 85, 13, 233, 798, 1227, 1084, 567, 166, 21, 610, 2362, 4230, 4510, 3038, 1286, 314, 34, 1597, 6898, 14058, 17474, 14284, 7814, 2774, 582, 55

Offset: 0

Philippe Deléham, Dec 18 2011

T(n,n) = Fibonacci(n+2) = A000045(n+2).

			Triangle begins :
1
2, 2
5, 8, 3
13, 27, 19, 5
34, 86, 86, 42, 8
89, 265, 338, 234, 85, 13

Cf. A000045, A001519, A122367, A000302 (row sums), A202395

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) + T(n-2,k-2) - T(n-2,k) with T(0,0) = 1, T(1,0) = T(1,1) = 2 and T(n,k) = 0 if k<0 or if n

G.f.: (1+(y-1)*x)/(1-(3+y)*x+(1-y^2)*x^2).

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A122367(n), A000302(n), A180035(n) for x = -1, 0, 1, 2 respectively.

Sum_{k, 0<=k<=n} T(n,k)*3^k = 2^n * A055099(n). - Philippe Deléham, Feb 05 2012

A276472 Modified Pascal's triangle read by rows: T(n,k) = T(n-1,k) + T(n-1,k-1), 12. T(n,n) = T(n,n-1) + T(n-1,n-1), n>1. T(1,1) = 1, T(2,1) = 1. n>=1.

Original entry on oeis.org

1, 1, 2, 4, 3, 5, 11, 7, 8, 13, 29, 18, 15, 21, 34, 76, 47, 33, 36, 55, 89, 199, 123, 80, 69, 91, 144, 233, 521, 322, 203, 149, 160, 235, 377, 610, 1364, 843, 525, 352, 309, 395, 612, 987, 1597, 3571, 2207, 1368, 877, 661, 704, 1007, 1599, 2584, 4181

Offset: 1

Yuriy Sibirmovsky, Sep 12 2016

The recurrence relations for the border terms are the only way in which this differs from Pascal's triangle.
Column T(2n,n+1) appears to be divisible by 4 for n>=2; T(2n-1,n) divisible by 3 for n>=2; T(2n,n-2) divisible by 2 for n>=3.
The symmetry of T(n,k) can be observed in a hexagonal arrangement (see the links).
Consider T(n,k) mod 3 = q. Terms with q = 0 show reflection symmetry with respect to the central column T(2n-1,n), while q = 1 and q = 2 are mirror images of each other (see the link).

			Triangle T(n,k) begins:
n\k 1    2    3    4   5    6    7    8    9
1   1
2   1    2
3   4    3    5
4   11   7    8    13
5   29   18   15   21   34
6   76   47   33   36   55   89
7   199  123  80   69   91   144 233
8   521  322  203  149  160  235 377  610
9   1364 843  525  352  309  395 612  987  1597
...
In another format:
__________________1__________________
_______________1_____2_______________
____________4_____3_____5____________
________11_____7_____8_____13________
____29_____18_____15____21_____34____
_76_____47____33_____36____55_____89_

Yuriy Sibirmovsky, T(n,k), read by rows as a linear sequence a(j) for j = 1..5050
Yuriy Sibirmovsky, Symmetrical hexagonal arrangement for initial terms of T(n,k)
Yuriy Sibirmovsky, T(n,k) compared with Pascal's triangle
Yuriy Sibirmovsky, Illustration for T(n,k) mod 3

Cf. A000045, A000204, A001519, A001906, A002878, A005248, A054441, A088305, A122367, A258109, A026671, A026726, A026732, A004187, A049685, A033891, A206351, A092521, A081018, A049684, A058038, A081016, A003482, A049629, A133273, A049664, A049683, A119032, A023039, A007805, A033889.

Nm=12;
T=Table[0,{n,1,Nm},{k,1,n}];
T[[1,1]]=1;
T[[2,1]]=1;
T[[2,2]]=2;
Do[T[[n,1]]=T[[n-1,1]]+T[[n,2]];
T[[n,n]]=T[[n-1,n-1]]+T[[n,n-1]];
If[k!=1&&k!=n,T[[n,k]]=T[[n-1,k]]+T[[n-1,k-1]]],{n,3,Nm},{k,1,n}];
{Row[#,"\t"]}&/@T//Grid

T(n,k) = if (k==1, if (n==1, 1, if (n==2, 1, T(n-1,1) + T(n,2))), if (kMichel Marcus, Sep 14 2016

Conjectures:
Relations with other sequences:
T(n+1,1) = A002878(n-1), n>=1.
T(n,n) = A001519(n) = A122367(n-1), n>=1.
T(n+1,2) = A005248(n-1), n>=1.
T(n+1,n) = A001906(n) = A088305(n), n>=1.
T(2n-1,n) = 3*A054441(n-1), n>=2. [the central column].
Sum_{k=1..n} T(n,k) = 3*A105693(n-1), n>=2. [row sums].
Sum_{k=1..n} T(n,k)-T(n,1)-T(n,n) = 3*A258109(n), n>=2.
T(2n,n+1) - T(2n,n) = A026671(n), n>=1.
T(2n,n-1) - T(2n,n) = 2*A026726(n-1), n>=2.
T(n,ceiling(n/2)) - T(n-1,floor(n/2)) = 2*A026732(n-3), n>=3.
T(2n+1,2n) = 3*A004187(n), n>=1.
T(2n+1,2) = 3*A049685(n-1), n>=1.
T(2n+1,2n) + T(2n+1,2) = 3*A033891(n-1), n>=1.
T(2n+1,3) = 5*A206351(n), n>=1.
T(2n+1,2n)/3 - T(2n+1,3)/5 = 4*A092521(n-1), n>=2.
T(2n,1) = 1 + 5*A081018(n-1), n>=1.
T(2n,2) = 2 + 5*A049684(n-1), n>=1.
T(2n+1,2) = 3 + 5*A058038(n-1), n>=1.
T(2n,3) = 3 + 5*A081016(n-2), n>=2.
T(2n+1,1) = 4 + 5*A003482(n-1), n>=1.
T(3n,1) = 4*A049629(n-1), n>=1.
T(3n,1) = 4 + 8*A119032(n), n>=1.
T(3n+1,3) = 8*A133273(n), n>=1.
T(3n+2,3n+2) = 2 + 32*A049664(n), n>=1.
T(3n,3n-2) = 4 + 32*A049664(n-1), n>=1.
T(3n+2,2) = 2 + 16*A049683(n), n>=1.
T(3n+2,2) = 2*A023039(n), n>=1.
T(2n-1,2n-1) = A033889(n-1), n>=1.
T(3n-1,3n-1) = 2*A007805(n-1), n>=1.
T(5n-1,1) = 11*A097842(n-1), n>=1.
T(4n+5,3) - T(4n+1,3) = 15*A000045(8n+1), n>=1.
T(5n+4,3) - T(5n-1,3) = 11*A000204(10n-2), n>=1.
Relations between left and right sides:
T(n,1) = T(n,n) - T(n-2,n-2), n>=3.
T(n,2) = T(n,n-1) - T(n-2,n-3), n>=4.
T(n,1) + T(n,n) = 3*T(n,n-1), n>=2.

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A208343 Triangle of coefficients of polynomials v(n,x) jointly generated with A208342; see the Formula section.

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A122392 Dimension of 3-variable non-commutative harmonics (Hausdorff derivative). The dimension of the space of non-commutative polynomials in 3 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( w ) = sum over all subwords of w deleting xi once).

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A373639 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-5*k,k).

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A155110 a(n) = 8*Fibonacci(2n+1).

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A210215 Triangle of coefficients of polynomials u(n,x) jointly generated with A210216; see the Formula section.

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A052568 E.g.f.: (1-x)/(1-3*x+x^2).

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A104726 Triangle generated as the matrix product of A026729 and A000012 (triangular views), read by rows.

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A164267 A Fibonacci convolution.

A373639 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-5k,k).