A003946 -id:A003946 - cp's OEIS frontend

A202337 Range of A062723.

1, 2, 6, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, 6377292, 19131876, 57395628, 172186884, 516560652, 1549681956, 4649045868, 13947137604, 41841412812, 125524238436, 376572715308, 1129718145924, 3389154437772, 10167463313316

Offset: 1

Reinhard Zumkeller, Dec 17 2011

nonn

Subsequence of A000792.

Apparently a(n) = A052156(n - 1) for n >= 4. - Georg Fischer, Mar 26 2019

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3).

Cf. A000792, A003946, A027327, A025579, A052156, A062723.

a202337 n = a202337_list !! (n-1)
a202337_list = f a062723_list where
   f (x:xs'@(x':xs)) = if x == x' then f xs' else x : f xs'

From Colin Barker, Mar 26 2019: (Start)
G.f.: x*(1 - x - 6*x^3) / (1 - 3*x).
a(n) = 4*3^(n-3) for n>3.
a(n) = 3*a(n-1) for n>4.
(End)

A208333 Triangle of coefficients of polynomials v(n,x) jointly generated with A208332; see the Formula section.

Original entry on oeis.org

1, 0, 4, 0, 2, 10, 0, 2, 6, 28, 0, 2, 6, 24, 76, 0, 2, 6, 28, 80, 208, 0, 2, 6, 32, 100, 264, 568, 0, 2, 6, 36, 120, 360, 840, 1552, 0, 2, 6, 40, 140, 464, 1232, 2624, 4240, 0, 2, 6, 44, 160, 576, 1680, 4128, 8064, 11584, 0, 2, 6, 48, 180, 696, 2184, 5952

Offset: 1

Clark Kimberling, Feb 26 2012

As triangle T(n,k) with 0 <= k <= n, it is (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 28 2012

			First five rows:
  1;
  0,  4;
  0,  2, 10;
  0,  2,  6, 28;
  0,  2,  6, 24, 76;
First five polynomials u(n,x):
  1
      4x
      2x + 10x^2
      2x +  6x^2 + 28x^3
      2x +  6x^2 + 24x^3 + 76x^4.

Cf. A003946, A026150, A208332.

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_] := 2 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[%]  (* A208332 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208333 *)

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = 2x*u(n-1,x) + 2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Feb 28 2012: (Start)
As triangle with 0 <= k <= n:
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 2*T(n-2,k-2) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 4 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1-x+2*y*x)/(1-x-2*y*x+2*y*x^2-2*y^2*x^2).
T(n,n) = A026150(n+1).
Sum_{k=0..n} T(n,k) = A003946(n). (End)

A294090 Base-10 complementary numbers: n equals the product of the 10's complement of its digits.

Original entry on oeis.org

5, 18, 35, 50, 180, 315, 350, 500, 1800, 3150, 3500, 5000, 18000, 31500, 35000, 50000, 180000, 315000, 350000, 500000, 1800000, 3150000, 3500000, 5000000, 18000000, 31500000, 35000000, 50000000, 180000000, 315000000, 350000000, 500000000, 1800000000

Offset: 1

M. F. Hasler, Feb 09 2018

The only primitive terms of the sequence, i.e., not equal to 10 times a smaller term, are 5, 18, 35 and 315.
For base 2, 3, 4 and 5, the corresponding sequences are less interesting: b = 2 yields powers of 2, A000079; b = 3 yields 4 times powers of 3, A003946 \ {1}; b = 4 yields {2, 6}*{4^k, k>=0} = A122756 = 2*A084221; b = 5 yields 8*{5^k, k>=0} = A128625 \ {1}.
See A298976 for base-6 complementary numbers. Base 7 yields {12, 120}*{7^k, k>=0}, cf. A298977. The linked web page (in French) gives also examples for base-100 complementary numbers, e.g., 198 = (100 - 1)*(100 - 98), 1680 = (100 - 16)*(100 - 80), ..., and for base-1000 complementary numbers.

			5 = (10-5), therefore 5 is in the sequence.
18 = (10-1)*(10-8), therefore 18 is in the sequence.
35 = (10-3)*(10-5), therefore 35 is in the sequence.
315 = (10-3)*(10-1)*(10-5), therefore 315 is in the sequence.
If x is in the sequence, then 10*x = concat(x,0) = x*(10-0) is in the sequence.

Colin Barker, Table of n, a(n) for n = 1..1000
G. Villemin, Nombres complémentés (in French).
Index entries for linear recurrences with constant coefficients, signature (0,0,0,10).

Cf. A298976, A298977.

LinearRecurrence[{0,0,0,10},{5,18,35,50,180,315},40] (* Harvey P. Dale, Mar 02 2024 *)

is(n,b=10)={n==prod(i=1,#n=digits(n,b),b-n[i])}

a(n)=if(n>6,a((n-3)%4+3)*10^((n-3)\4),[5,18,35,50,180,315][n])

Vec(x*(5 + 18*x + 35*x^2 + 50*x^3 + 130*x^4 + 135*x^5) / (1 - 10*x^4) + O(x^60)) \\ Colin Barker, Feb 09 2018

a(n+4) = 10 a(n) for all n >= 3.

G.f.: x*(5 + 18*x + 35*x^2 + 50*x^3 + 130*x^4 + 135*x^5) / (1 - 10*x^4). - Colin Barker, Feb 09 2018

A027327 a(n) = Sum_{k=0..m} (k+1) * A026120(n, m-k), where m=0 for n=0,1; m=n for n >= 2.

Original entry on oeis.org

1, 3, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, 6377292, 19131876, 57395628, 172186884, 516560652, 1549681956, 4649045868, 13947137604, 41841412812, 125524238436, 376572715308

Offset: 0

Clark Kimberling

nonn

Index entries for linear recurrences with constant coefficients, signature (3).

Join[{1,3},NestList[3#&,12,30]] (* Harvey P. Dale, Aug 13 2024 *)

G.f.: (1+3x^2)/(1-3x). - Paul Barry, May 19 2003

a(n) = A003946(n), n>1. - R. J. Mathar, Oct 23 2008

A162740 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

Original entry on oeis.org

1, 4, 12, 30, 72, 168, 390, 900, 2076, 4782, 11016, 25368, 58422, 134532, 309804, 713406, 1642824, 3783048, 8711526, 20060676, 46195260, 106377294, 244963080, 564094968, 1298984214, 2991269124, 6888221772, 15862029150, 36526694472, 84112781928, 193692865350

Offset: 0

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

The initial terms coincide with those of A003946, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.
From Bruno Berselli, Dec 28 2015: (Start)
Also, expansion of b(2)*b(3)/(1 - 2*x - 2*x^2 + 3*x^3), where b(k) = (1-x^k)/(1-x).
This is also the Poincaré series [or Poincare series] for the quasi-Lannér diagram QL4_22 - see Table 7.8 in Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2009), or equivalently Table 6 in the shorter version, Maxim Chapovalov, Dimitry Leites and Rafael Stekolshchik (2010).
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, arXiv:0906.1596 [math.RT], 2009, page 31.
Maxim Chapovalov, Dimitry Leites, and Rafael Stekolshchik, The Poincaré series [or Poincare series] of the hyperbolic Coxeter groups with finite volume of fundamental domains, Journal of Nonlinear Mathematical Physics 17.supp01 (2010), page 186.
Index entries for linear recurrences with constant coefficients, signature (2,2,-3).

Cf. similar sequences listed in A265055.

m:=40; R:=PowerSeriesRing(Integers(), m); b:=func; Coefficients(R!(b(2)*b(3)/(1-2*x-2*x^2+3*x^3))); // Bruno Berselli, Dec 28 2015 - see Chapovalov et al.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^3)/(1-3*x+5*x^3-3*x^4) )); // G. C. Greubel, Apr 25 2019

CoefficientList[Series[(x^3+2x^2+2x+1)/(3x^3-2x^2-2x+1), {x, 0, 40}], x ] (* Vincenzo Librandi, Apr 29 2014 *)
coxG[{3, 3, -2, 40}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 25 2019 *)

my(x='x+O('x^40)); Vec((1+x)*(1-x^3)/(1-3*x+5*x^3-3*x^4)) \\ G. C. Greubel, Apr 25 2019

((1+x)*(1-x^3)/(1-3*x+5*x^3-3*x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 25 2019

G.f.: (x^3 + 2*x^2 + 2*x + 1)/(3*x^3 - 2*x^2 - 2*x + 1).
From Bruno Berselli, Dec 28 2015: (Start)
a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) for n>3.
a(n) = -2 + ((-7+2*sqrt(13))*(1-sqrt(13))^n + (7+2*sqrt(13))*(1+sqrt(13))^n)/(3*sqrt(13)*2^(n-1)) for n>0. (End)
G.f.: (1+x)*(1-x^3)/(1 -3*x +5*x^3 -3*x^4). - G. C. Greubel, Apr 25 2019

A166468 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

Original entry on oeis.org

1, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708582, 2125728, 6377136, 19131264, 57393360, 172178784, 516532464, 1549585728, 4648722192, 13946061600, 41837869872, 125512664832, 376535160174, 1129596977628

Offset: 0

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

The initial terms coincide with those of A003946, although the two sequences are eventually different.

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

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

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^12)/(1-3*x+5*x^12-3*x^13) )); // G. C. Greubel, Apr 26 2019

CoefficientList[Series[(1+x)*(1-x^12)/(1 -3*x +5*x^12 -3*x^13), {x, 0, 30}], x ] (* Vincenzo Librandi, Apr 29 2014 *)(* modified by G. C. Greubel, Apr 26 2019 *)
coxG[{12,3,-2,30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, May 09 2018 *)

my(x='x+O('x^30)); Vec((1+x)*(1-x^12)/(1-3*x+5*x^12-3*x^13)) \\ G. C. Greubel, Apr 26 2019

((1+x)*(1-x^12)/(1-3*x+5*x^12-3*x^13)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019

G.f.: (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)/(3*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).
G.f.: (1+x)*(1-x^12)/(1 -3*x +5*x^12 -3*x^13). - G. C. Greubel, Apr 26 2019
a(n) = -3*a(n-12) + 2*Sum_{k=1..11} a(n-k). - Wesley Ivan Hurt, May 06 2021

A182950 Joint-rank array of the numbers (3i+2)3^j, where i>=0, j>=0, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 9, 7, 4, 27, 22, 12, 5, 81, 67, 36, 16, 6, 243, 202, 108, 49, 20, 8, 729, 607, 324, 148, 62, 25, 10, 2187, 1822, 972, 445, 188, 76, 30, 11, 6561, 5467, 2916, 1336, 566, 229, 90, 34, 13, 19683, 16402, 8748, 4009, 1700, 688, 270, 103, 39, 14

Offset: 1

Clark Kimberling, Dec 15 2010

Joint-rank arrays are defined in the first comment at A182801.  As for any joint-rank array, A182950 is a permutation of the positive integers, but, a fortiori, A182950 is an interspersion: after initial terms every row is interspersed with all other rows.  The numbers (3*i+2)*3^j as an array comprise A182830; and sorted, possibly A026179.
(row 1)=A000244.
(row 2)=A060816.
(row 3)=A003946.
(row 4)=A052909.
(row 5)=A027107?

			Northwest corner:
1....3....9....27...
2....7...22....67...
4...12...36...108...
5...16...49...148...

Clark Kimberling, Interspersions and Dispersions.

Cf. A182801, A182830, A026179, A182949.

 M[i_,j_]:=j+Floor[Log[3*i/2+1]/Log[3]];
 T[i_,j_]:=Sum[Floor[1/3+(3*i+2)*3^(j-k-1)],{k,0,M[i,j]}];
 TableForm[Table[T[i,j],{i,0,9},{j,0,9}]]

A344122 Triangle T(n,k) read by rows in which n-th row gives all tree-able degree sequences S of n nodes encoded as Product_{k in S} prime(k); n >= 2, 1<= k <= A000041(n-2).

Original entry on oeis.org

4, 12, 40, 36, 112, 120, 108, 352, 336, 400, 360, 324, 832, 1056, 1120, 1008, 1200, 1080, 972, 2176, 2496, 3520, 3136, 3168, 3360, 4000, 3024, 3600, 3240, 2916, 4864, 6528, 8320, 9856, 7488, 10560, 9408, 11200, 9504, 10080, 12000, 9072, 10800, 9720, 8748, 11776, 14592, 21760

Offset: 2

Washington Bomfim, Jun 02 2021

Tree-able degree sequences are degree sequences that can be realized as trees [Stern].

The partitions of n-2 are given in nondecreasing order of length/lex.

			Triangle T(n,k) begins:
n/k 1    2     3 ...
2     4;
3    12;
4    40,   36;
5   112,  120,  108;
6   352,  336,  400,  360,  324;
7   832, 1056, 1120, 1008, 1200, 1080,  972;
8  2176, 2496, 3520, 3136, 3168, 3360, 4000, 3024, 3600, 3240, 2916;
  ...
Row 5 is 112, 120, 108 because prime(1) = 2, prime(2) = 3, prime(3) = 5, and prime(4) = 7. The tree-able degree sequences of 5 nodes, related tree realization and encode are given below.
  [4, 1, 1, 1, 1]     o           7*2*2*2*2 = 112.
                   ( ) ( )
                   o o o o
  [3, 2, 1, 1, 1]        o        5*3*2*2*2 = 120.
                       / | \
                   o--o  o  o
  [2, 2, 2, 1, 1]  o--o--o--o--o  3*3*3*2*2 = 108.

Washington Bomfim, Table of n, a(n) for n = 2..9297 (Rows n = 2..27, flattened)
Samuel Stern, The Tree of Trees: on methods for finding all non-isomorphic tree-realizations of degree sequences, Honors Thesis, Wesleyan University, 2017.
Index entries for sequences computed from indices in prime factorization

Cf. A000041, A000055, A003946 (last terms in rows), A215366, A265127 (first column).

\\ Gives row n of triangle, n >= 2.
Row(n)={my(j=1, V=vector(numbpart(n-2))); forpart(P=n-2,
V[j] = prod(k = 1, #P, prime(P[k] + 1)); V[j] <<= (n-#P); j++ ); V };

A128932 Define the Fibonacci polynomials by F[1] = 1, F[2] = x; for n > 2, F[n] = xF[n-1] + F[n-2] (cf. A049310, A053119). Swamy's inequality implies that F[n] <= F[n]^2 <= G[n] = (x^2 + 1)^2(x^2 + 2)^(n-3) for n >= 3 and x >= 1. The sequence gives a triangle of coefficients of G[n] - F[n] read by rows.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, -2, 5, -1, 4, 0, 1, 3, 0, 9, 0, 12, 0, 6, 0, 1, 8, -3, 28, -4, 38, -1, 25, 0, 8, 0, 1, 15, 0, 58, 0, 99, 0, 87, 0, 41, 0, 10, 0, 1, 32, -4, 144, -10, 272, -6, 280, -1, 170, 0, 61, 0, 12, 0, 1, 63, 0, 310, 0, 673, 0, 825, 0, 619, 0, 292, 0, 85, 0, 14, 0, 1

Offset: 3

N. J. A. Sloane, Apr 28 2007

From Petros Hadjicostas, Jun 10 2020: (Start)
Swamy's (1966) inequality states that F[n]^2 <= G[n] for all real x and all integers n >= 3. Because F[n] >= 1 for all real x >= 1, we get F[n] <= G[n] for all integers n >= 3 and all real x >= 1.
Row n >= 3 of this irregular table gives the coefficients of the polynomial G[n] - F[n] (with exponents in increasing order). The degree of G[n] - F[n] is 2*n - 2, so row n >= 3 contains 2*n - 1 terms.
Guilfoyle (1967) notes that F[n] = det(A_n), where A_n is the (n-1) X (n-1) matrix [[x, -1, 0, 0, ..., 0, 0, 0], [1, x, -1, 0, ..., 0, 0, 0], [0, 1, x, -1, ..., 0, 0, 0], ..., [0, 0, 0, 0, ..., 1, x, -1], [0, 0, 0, 0, ..., 0, 1, x]], and Swamy's original inequality follows from Hadamard's inequality.
Koshy (2019) writes Swamy's original inequality in the form x^(n-3)*F[n]^2 <= F[3]^2*F[4]^(n-3) for x >= 1, and gives a counterpart inequality for Lucas polynomials. Notice, however, that the original form of Swamy's inequality is true for all real x. (End)

			Triangle T(n,k) (with rows n >= 3 and columns k = 0..2*n-2) begins:
   0,  0,  1,  0,  1;
   2, -2,  5, -1,  4,  0,  1;
   3,  0,  9,  0, 12,  0,  6, 0,  1;
   8, -3, 28, -4, 38, -1, 25, 0,  8, 0,  1;
  15,  0, 58,  0, 99,  0, 87, 0, 41, 0, 10, 0, 1;
  ...

Thomas Koshy, Fibonacci and Lucas numbers with Applications, Vol. 2, Wiley, 2019; see p. 33. [Vol. 1 was published in 2001.]
D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, 1970; p. 232, Sect. 3.3.38.

Richard Guilfoyle, Comment to the solution of Problem E1846, Amer. Math. Monthly, 74(5), 1967, 593. [It is pointed out that the inequality is a special case of Hadamard's inequality.]
M. N. S. Swamy, Problem E1846 proposed for solution, Amer. Math. Monthly, 73(1) (1966), 81.
M. N. S. Swamy and R. E. Giudici, Solution to Problem E1846, Amer. Math. Monthly, 74(5), 1967, 592-593.
M. N. S. Swamy and Joel Pitcain, Comment to Problem E1846, Amer. Math. Monthly, 75(3) (1968), 295. [It is pointed out that I^{n-1}*F[n](x) = U_{n-1}(I*x/2), where U_{n-1}(cos(t)) = sin(n*t)/sin(t) and I = sqrt(-1); Cf. A049310 and A053119, but with different notation.]
Wikipedia, Fibonacci polynomials.
Wikipedia, Hadamard's inequality.

Cf. A000045, A003946, A049310, A053119, A166920.

lista(nn) = {my(f=vector(nn)); my(g=vector(nn)); my(h=vector(nn)); f[1]=1; f[2]=x; g[1]=0; g[2]=0; for(n=3, nn, g[n] = (x^2+1)^2*(x^2+2)^(n-3)); for(n=3, nn, f[n] = x*f[n-1]+f[n-2]); for(n=1, nn, h[n] = g[n]-f[n]); for(n=3, nn, for(k=0, 2*n-2, print1(polcoef(h[n], k, x), ",")); print(););} \\ Petros Hadjicostas, Jun 10 2020

From Petros Hadjicostas, Jun 10 2020: (Start)
T(n,0) = 2^(n-3) - (1 - (-1)^n)/2 = A166920(n-3) for n >= 3.
Sum_{k=0}^{2*n-2} T(n,k) = 4*3^(n-3) - Fib(n) = A003946(n-2) - A000045(n) for n >= 3. (End)

Name edited by Petros Hadjicostas, Jun 10 2020

A143865 Eigentriangle of A099375 (odd number subsequences decrescendo).

Original entry on oeis.org

1, 3, 1, 5, 3, 4, 7, 5, 12, 12, 9, 7, 20, 36, 36, 11, 9, 28, 60, 108, 108, 13, 11, 36, 84, 180, 324, 324, 15, 13, 44, 108, 252, 540, 972, 972, 17, 15, 52, 132, 324, 756, 1620, 2916, 2916

Offset: 1

Gary W. Adamson, Sep 04 2008

The product A099375 * (A003946 * 0^(n-k)) creates a new eigentriangle with rows = termwise product of odd descrendo terms, e.g.: (7, 5, 3, 1) and first n terms of (1, 1, 4, 12,). The resulting triangle has row sums = (1, 4, 12, 36, 108,...), right border = (1, 1, 4, 12, 36,...) and the property that sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle = 1; 3, 1; 5, 3, 4; 7, 5, 12, 12; 9, 7, 20, 36, 36; 11, 9, 28, 60, 108, 108; 13, 11, 36, 84, 180, 324, 324; ... Row 4 = termwise product of (7, 5, 3, 1) and (1, 1, 4, 12) = (7*1, 5*1, 3*4, 1*12).

A099375, Cf. A003946

Triangle read by rows, A099375: (1; 3,1; 5,3,1;...) * (A003946 * 0^(n-k)); 1<=k<=n. (A003946 * 0^(n-k) = an infinite lower triangular matrix with shifted A003946: (1, 1, 4, 12, 36, 108,...) in the main diagonal and the rest zeros.

cp's OEIS Frontend

A202337 Range of A062723.

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A208333 Triangle of coefficients of polynomials v(n,x) jointly generated with A208332; see the Formula section.

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A294090 Base-10 complementary numbers: n equals the product of the 10's complement of its digits.

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A027327 a(n) = Sum_{k=0..m} (k+1) * A026120(n, m-k), where m=0 for n=0,1; m=n for n >= 2.

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A162740 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

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A166468 Number of reduced words of length n in Coxeter group on 4 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

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A182950 Joint-rank array of the numbers (3*i+2)*3^j, where i>=0, j>=0, by antidiagonals.

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A344122 Triangle T(n,k) read by rows in which n-th row gives all tree-able degree sequences S of n nodes encoded as Product_{k in S} prime(k); n >= 2, 1<= k <= A000041(n-2).

A182950 Joint-rank array of the numbers (3i+2)3^j, where i>=0, j>=0, by antidiagonals.