A003946 -id:A003946 - cp's OEIS frontend

A287831 Number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 8.

1, 10, 96, 924, 8892, 85572, 823500, 7924932, 76265388, 733938084, 7063035084, 67970944260, 654116708844, 6294876045156, 60578584659468, 582976518206148, 5610260171812140, 53990200655546148, 519573366930788172, 5000101506310370436, 48118353758378062956

Offset: 0

David Nacin, Jun 02 2017

In general, the number of sequences over the alphabet {0,1,...,9} such that no two consecutive terms have distance 5+k for k in {0,1,2,3,4} is given by a(n) = 9*a(n-1) + 2*k*a(n-2), a(0)=1, a(1)=10.

Index entries for linear recurrences with constant coefficients, signature (9,6).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287831.

Mathematica
```
LinearRecurrence[{9, 6}, {1, 10}, 30]
```

def a(n):
 if n in [0, 1]:
  return [1, 10][n]
 return 9*a(n-1)+6*a(n-2)

a(n) = 9*a(n-1) + 6*a(n-2), a(0)=1, a(1)=10.
G.f.: (-1 - x)/(-1 + 9*x + 6*x^2).
a(n) = ((1 - 11/sqrt(105))/2)*((9 - sqrt(105))/2)^n + ((1 + 11/sqrt(105))/2)*((9 + sqrt(105))/2)^n.

A326339 Number of connected simple graphs with vertices {1..n} and no crossing or nesting edges.

Original entry on oeis.org

1, 0, 1, 4, 12, 36, 108, 324

Offset: 0

Gus Wiseman, Jun 29 2019

Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d.

Appears to be essentially the same as A003946.

			The a(2) = 1 through a(4) = 36 edge-sets:
  {12}  {12,13}     {12,13,14}
        {12,23}     {12,13,34}
        {13,23}     {12,14,34}
        {12,13,23}  {12,23,24}
                    {12,23,34}
                    {12,24,34}
                    {13,23,34}
                    {14,24,34}
                    {12,13,14,34}
                    {12,13,23,34}
                    {12,14,24,34}
                    {12,23,24,34}

Gus Wiseman, The a(5) = 36 connected graphs with no crossing or nesting edges.
Gus Wiseman, The a(6) = 108 connected graphs with no crossing or nesting edges.

Covering graphs with no crossing or nesting edges are A326329.
Connected simple graphs are A001349.
The case with only crossing edges forbidden is A007297.
Graphs without crossing or nesting edges are A326244.
Cf. A006125, A054726, A117662, A136653.
Cf. A324169, A326210, A326293, A326340.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[csm[#]]<=1&&!MatchQ[#,{_,{x_,y_},_,{z_,t_},_}/;x

A025579 a(1)=1, a(2)=2, a(n) = 4*3^(n-3) for n >= 3.

Original entry on oeis.org

1, 2, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, 6377292, 19131876, 57395628, 172186884, 516560652, 1549681956, 4649045868, 13947137604, 41841412812, 125524238436, 376572715308, 1129718145924

Offset: 1

Clark Kimberling

a(n) is the sum of the numbers in row n+1 of the array defined in A025564 (and of the array in A024996).
a(n) is the number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3.
Equals binomial transform of A095342: (1, 1, 5, 5, 17, 25, 61, ...). - Gary W. Adamson, Mar 04 2010

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

Cf. A003946, A024996, A025564, A095342.

Concatenation([1,2], List([3..30], n-> 4*3^(n-3) )); # G. C. Greubel, Dec 26 2019

[1,2] cat [4*3^(n-3): n in [3..30]]; // G. C. Greubel, Dec 26 2019

seq( `if`(n<3, n, 4*3^(n-3)), n=1..30); # G. C. Greubel, Dec 26 2019

Join[{1,2},4*3^Range[0,30]] (* or *) Join[{1,2},NestList[3#&,4,30]] (* Harvey P. Dale, Jun 27 2011 *)

a(n)=max(n,4*3^(n-3)) \\ Charles R Greathouse IV, Jun 28 2011

Vec(x*(1+x)*(1-2*x)/(1-3*x) + O(x^30)) \\ Colin Barker, Oct 29 2019

[1,2]+[4*3^(n-3) for n in (3..30)] # G. C. Greubel, Dec 26 2019

a(n) = A003946(n-2), n>2. - R. J. Mathar, May 28 2008
From Colin Barker, Oct 29 2019: (Start)
G.f.: x*(1 + x)*(1 - 2*x) / (1 - 3*x).
a(n) = 3*a(n-1) for n>3. (End)

Definition corrected by R. J. Mathar, May 28 2008

A123620 Expansion of (1 + x + x^2) / (1 - 3x - 3x^2).

Original entry on oeis.org

1, 4, 16, 60, 228, 864, 3276, 12420, 47088, 178524, 676836, 2566080, 9728748, 36884484, 139839696, 530172540, 2010036708, 7620627744, 28891993356, 109537863300, 415289569968, 1574482299804, 5969315609316, 22631393727360, 85802128010028, 325300565212164

Offset: 0

N. J. A. Sloane, Nov 20 2006

From Johannes W. Meijer, Aug 14 2010: (Start)
A berserker sequence, see A180141. For the corner squares 16 A[5] vectors with decimal values between 3 and 384 lead to this sequence. These vectors lead for the side squares to A180142 and for the central square to A155116.
This sequence belongs to a family of sequences with GF(x) = (1+x+k*x^2)/(1-3*x+(k-4)*x^2). Berserker sequences that are members of this family are 4*A055099(n) (k=2; with leading 1 added), A123620 (k=1; this sequence), A000302 (k=0), 4*A179606 (k=-1; with leading 1 added) and A180141 (k=-2). Some other members of this family are 4*A003688 (k=3; with leading 1 added), 4*A003946 (k=4; with leading 1 added), 4*A002878 (k=5; with leading 1 added) and 4*A033484 (k=6; with leading 1 added).
(End)
a(n) is the number of length n sequences on an alphabet of 4 letters that do not contain more than 2 consecutive equal letters. For example, a(3)=60 because we count all 4^3=64 words except: aaa, bbb, ccc, ddd. - Geoffrey Critzer, Mar 12 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, arXiv:math/0112281 [math.CO], 2001.
A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 1-14.
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 205
Index entries for linear recurrences with constant coefficients, signature (3,3).

Column 4 in A265584.

[1] cat [Round(((2^(1-n)*(-(3-Sqrt(21))^(1+n) + (3+Sqrt(21))^(1+n))))/(3*Sqrt(21))): n in [1..50]]; // G. C. Greubel, Oct 26 2017

nn=25;CoefficientList[Series[(1-z^(m+1))/(1-r z +(r-1)z^(m+1))/.{r->4,m->2},{z,0,nn}],z] (* Geoffrey Critzer, Mar 12 2014 *)
CoefficientList[Series[(1 + x + x^2)/(1 - 3 x - 3 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 14 2014 *)
LinearRecurrence[{3,3},{1,4,16},30] (* Harvey P. Dale, Jul 14 2023 *)

my(x='x+O('x^50)); Vec((1+x+x^2)/(1-3*x-3*x^2)) \\ G. C. Greubel, Oct 16 2017

a(0)=1, a(1)=4, a(2)=16, a(n)=3*a(n-1)+3*a(n-2) for n>2. - Philippe Deléham, Sep 18 2009

a(n) = ((2^(1-n)*(-(3-sqrt(21))^(1+n) + (3+sqrt(21))^(1+n)))) / (3*sqrt(21)) for n>0. - Colin Barker, Oct 17 2017

A208904 Triangle of coefficients of polynomials v(n,x) jointly generated with A208660; see the Formula section.

Original entry on oeis.org

1, 3, 1, 5, 6, 1, 7, 19, 9, 1, 9, 44, 42, 12, 1, 11, 85, 138, 74, 15, 1, 13, 146, 363, 316, 115, 18, 1, 15, 231, 819, 1059, 605, 165, 21, 1, 17, 344, 1652, 2984, 2470, 1032, 224, 24, 1, 19, 489, 3060, 7380, 8378, 4974, 1624, 292, 27, 1, 21, 670, 5301, 16488

Offset: 1

Clark Kimberling, Mar 03 2012

For a discussion and guide to related arrays, see A208510.
Riordan array ((1+x)/(1-x)^2, x(1+x)/(1-x)^2) (follows from Kruchinin formula). - Ralf Stephan, Jan 02 2014
From Peter Bala, Jul 21 2014: (Start)
Let M denote the lower unit triangular array A099375 and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/
having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... (which is clearly well-defined). See the Example section. (End)

			First five rows:
1
3...1
5...6....1
7...19...9....1
9...44...42...12...1
First five polynomials v(n,x):
1
3 + x
5 + 6x + x^2
7 + 19x + 9x^2 + x^3
9 + 44x + 42x^2 + 12x^3 + x^4
From _Peter Bala_, Jul 21 2014: (Start)
With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins
/1        \/1        \/1        \      /1            \
|3 1      ||0 1      ||0 1      |      |3  1         |
|5 3 1    ||0 3 1    ||0 0 1    |... = |5  6  1      |
|7 5 3 1  ||0 5 3 1  ||0 0 3 1  |      |7 19  9  1   |
|9 7 5 3 1||0 7 5 3 1||0 0 5 3 1|      |9 44 42 12 1 |
|...      ||...      ||...      |      |...
(End)

Cf. A208660, A208510. A099375.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + (x + 1)*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[%]    (* A208660 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208904 *)

u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
From Vladimir Kruchinin, Mar 11 2013: (Start)
T(n,k) = sum(i=0..n, binomial(i+k-1,2*k-1)*binomial(k,n-i))
((x+x^2)/(1-x)^2)^k = sum(n>=k, T(n,k)*x^n).
T(n,2)=A005900(n).
T(2*n-1,n) / n = A003169(n).
T(2*n,n) = A156894(n), n>1.
sum(k=1..n, T(n,k)) = A003946(n).
sum(k=1..n, T(n,k)*(-1)^(n+k)) = A078050(n).
n*sum(k=1..n, T(n,k)/k) = A058481(n). (End)
Recurrence: T(n+1,k+1) = sum {i = 0..n-k} (2*i + 1)*T(n-i,k). - Peter Bala, Jul 21 2014

A023693 Numbers with exactly 2 1's in ternary expansion.

Original entry on oeis.org

4, 10, 12, 14, 16, 22, 28, 30, 32, 34, 36, 38, 42, 44, 46, 48, 50, 52, 58, 64, 66, 68, 70, 76, 82, 84, 86, 88, 90, 92, 96, 98, 100, 102, 104, 106, 108, 110, 114, 116, 126, 128, 132, 134, 136, 138, 140, 142, 144, 146, 150, 152, 154, 156, 158, 160, 166, 172, 174

Offset: 1

Olivier Gérard

From Bernard Schott, Jun 19 2023: (Start)
A003946 \ {1} is a subsequence since for m = 4*3^(k-1) with k >= 1, ternary expansion of m is 110...0 with (k-1) trailing 0's.
A034472 \ {2} is a subsequence since for m = 3^k + 1 with k >= 1, ternary expansion of m is 10...01 with (k-1) 0's between first and last 1's. (End)

			30 is a term since 30 = 1010_3 with exactly 2 1's.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003946, A007089, A034472.

Select[ Range[ 162 ], (Count[ IntegerDigits[ #, 3 ], 1 ]==2)& ]

isok(k) = #select(x->(x==1), digits(k, 3)) == 2; \\ Michel Marcus, Jun 19 2023

A208340 Triangle of coefficients of polynomials v(n,x) jointly generated with A202390; see the Formula section.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 13, 14, 5, 5, 24, 41, 30, 8, 6, 40, 96, 109, 60, 13, 7, 62, 196, 308, 262, 116, 21, 8, 91, 364, 743, 868, 590, 218, 34, 9, 128, 630, 1604, 2413, 2240, 1267, 402, 55, 10, 174, 1032, 3186, 5926, 7046, 5424, 2627, 730, 89, 11, 230, 1617

Offset: 1

Clark Kimberling, Feb 27 2012

v(n,n) = F(n+1), where F=A000045, the Fibonacci numbers.
Alternating row sums of v: (1,0,0,0,0,0,0,0,...).
As triangle T(n,k) with 0 <= k <= n, it is (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.

			First five rows:
  1;
  2,  2;
  3,  6,  3;
  4, 13, 14,  5;
  5, 24, 41, 30,  8;
The first five polynomials v(n,x):
  1
  2 +  2x
  3 +  6x +  3x^2
  4 + 13x + 14x^2 +  5x^3
  5 + 24x + 41x^2 + 30x^3 + 8x^4

Cf. A202390.

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 + 1)*u[n - 1, x] + (x + 1)*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[%]    (* A202390 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208340 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* u row sums *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* v row sums *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* u alt. row sums *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* v alt. row sums *)

u(n,x) = u(n-1,x) + x*v(n-1,x), v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x), where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Feb 28 2012: (Start)
As triangle T(n,k) with 0 <= k <= n:
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-2) with T(0,0) = 1, T(1,0) = T(1,1) = 2 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1+y*x)/(1-2*x-y*x+x^2-y^2*x^2).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000027(n+1), A003946(n), A109115(n), A180031(n) for x = -1, 0, 1, 2, 3 respectively. (End)

A100852 Triangle read by rows: T(n,k) = 2^k * 3^n, 0 <= k <= n.

Original entry on oeis.org

1, 3, 6, 9, 18, 36, 27, 54, 108, 216, 81, 162, 324, 648, 1296, 243, 486, 972, 1944, 3888, 7776, 729, 1458, 2916, 5832, 11664, 23328, 46656, 2187, 4374, 8748, 17496, 34992, 69984, 139968, 279936, 6561, 13122, 26244, 52488, 104976, 209952, 419904, 839808

Offset: 0

Reinhard Zumkeller, Nov 20 2004

T(n,0) = A000244(n); T(n,n) = A000400(n) = A100851(n,n);
T(n,1) = A008776(n) for n>0;
T(n,2) = A003946(n+1) for n>1;
T(n,3) = A005051(n+1) for n>2;
T(n,n-1) = A081341(n+1) for n>0;
row sums give A016137.

			Triangle begins:
   1;
   3,   6;
   9,  18,  36;
  27,  54, 108, 216;
  81, 162, 324, 648, 1296;
...

Michael De Vlieger, Table of n, a(n) for n = 0..10010 (Rows 0 <= n <= 140).
Eric Weisstein's World of Mathematics, Smooth Number

Cf. A100851, A003586, A065333(T(n, k))=1.

Table[2^k*3^n, {n, 0, 140}, {k, 0, n}] // Flatten (* Michael De Vlieger, Mar 06 2017 *)

for(n=0, 8, for(k=0, n, print1(2^k*3^n", "))) \\ Satish Bysany, Mar 06 2017

G.f.: 1/((1 - 3*x)(1 - 6*x*y)). - Ilya Gutkovskiy, Jun 03 2017

A113071 Expansion of g.f. ((1+x)/(1-3*x))^2.

Original entry on oeis.org

1, 8, 40, 168, 648, 2376, 8424, 29160, 99144, 332424, 1102248, 3621672, 11809800, 38263752, 123294312, 395392104, 1262703816, 4017693960, 12741829416, 40291730856, 127073920392, 399817944648, 1255242384360, 3933092804328

Offset: 0

Paul Barry, Oct 14 2005

Binomial transform is A014916. In general, ((1+x)/(1-r*x))^2 expands to a(n) = ((r+1)*r^n*((r+1)*n + r-1) + 0^n)/r^2, which is also a(n) = Sum_{k=0..n} C(n,k)*Sum_{j=0..k} (j+1)*(r+1)^j. This is the self-convolution of the coordination sequence for the infinite tree with valency r.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-9).

Cf. A003946, A014916, A081038.

a:=[1,8,40];; for n in [4..30] do a[n]:=6*a[n-1]-9*a[n-2]; od; a; # G. C. Greubel, May 24 2019

I:=[8,40]; [1] cat [n le 2 select I[n] else 6*Self(n-1) - 9*Self(n-2): n in [1..30]]; // G. C. Greubel, May 24 2019

CoefficientList[Series[(1+x)^2/(1-3x)^2, {x, 0, 30}], x] (* Georg Fischer, May 24 2019 *)
LinearRecurrence[{6,-9}, {1,8,40}, 30] (* G. C. Greubel, May 24 2019 *)

my(x='x+O('x^30)); Vec(((1+x)/(1-3*x))^2) \\ G. C. Greubel, May 24 2019

(((1+x)/(1-3*x))^2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019

G.f.: (1+x)^2/(1-3*x)^2. [Corrected by Georg Fischer, May 24 2019]
a(n) = (8*3^n*(2*n+1) + 0^n)/9 = (4*3^n*(4*n+2) + 0^n)/9;
a(n) = Sum_{k=0..n} A003946(k)*A003946(n-k).
a(n) = Sum_{k=0..n} C(n, k)*Sum_{j=0..k} (j+1)*4^j.
a(n) = 8*A081038(n-1), n>0. - R. J. Mathar, Nov 28 2014
E.g.f.: (1 + 8*exp(3*x)*(1 + 6*x))/9. - Stefano Spezia, Jan 31 2025

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

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