A038208 -id:A038208 - cp's OEIS frontend

A257618 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = 8*x + 2.

1, 2, 2, 4, 40, 4, 8, 472, 472, 8, 16, 4928, 16992, 4928, 16, 32, 49824, 433984, 433984, 49824, 32, 64, 499584, 9505728, 22567168, 9505728, 499584, 64, 128, 4999040, 192085632, 909941120, 909941120, 192085632, 4999040, 128

Offset: 0

Dale Gerdemann, May 09 2015

			Triangle begins as:
    1;
    2,       2;
    4,      40,         4;
    8,     472,       472,         8;
   16,    4928,     16992,      4928,        16;
   32,   49824,    433984,    433984,     49824,        32;
   64,  499584,   9505728,  22567168,   9505728,    499584,      64;
  128, 4999040, 192085632, 909941120, 909941120, 192085632, 4999040, 128;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000079, A144828 (row sums), A167884.
Cf. A038208, A256890, A257609, A257610, A257612, A257614, A257616, A257617, A257619.
Similar sequences listed in A256890.

T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,8,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)

def T(n,k,a,b): # A257618
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
flatten([[T(n,k,8,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.
Sum_{k=0..n} T(n, k) = A144828(n).
From G. C. Greubel, Mar 24 2022: (Start)
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 8, and b = 2.
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n).
T(n, 1) = 2^(n-1)*(5^n - 2*n - 1).
T(n, 2) = 2^(n-3)*(3^(2*n+1) -2*(2*n+1)*5^n -1 +4*n^2). (End)

A300184 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (2x + 2)^n + (x^2 - 1)(x + 2)^n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 4, 7, 4, 1, 0, 12, 26, 19, 6, 1, 0, 32, 88, 88, 39, 8, 1, 0, 80, 272, 360, 230, 71, 10, 1, 0, 192, 784, 1312, 1140, 532, 123, 12, 1, 0, 448, 2144, 4368, 4872, 3164, 1162, 211, 14, 1, 0, 1024, 5632, 13568, 18592, 15680, 8176, 2480, 367, 16, 1, 0, 2304, 14336, 39936, 65088, 67872, 46368, 20304, 5262, 655, 18, 1

Offset: 0

Franck Maminirina Ramaharo, Feb 27 2018

Let L(n;x) = x*(2*x + 2)^n. Then T(n,k) is obtained from the expansion of the polynomial P(n;x) = (x + 2)*P(n-1;x) + L(n-1;x), with P(0;x) = x^2.

Let an n-chain link be the planar diagram that consists of n unknotted circles, linked together in a closed chain. Then T(n,k) is the number of diagrams having k components that are obtained by smoothing each double point (crossing). Kauffman defines the 'smoothing' of a framed 4-graph at a vertex v as "any of the two framed 4-graphs obtained by removing v and repasting the edges" (see links).

			The triangle T(n,k) begins:
n\k  0     1      2      3      4      5      6      7     8    9  10 11
0:   0     0      1
1:   0     1      2      1
2:   0     4      7      4      1
3:   0    12     26     19      6      1
4:   0    32     88     88     39      8      1
5:   0    80    272    360    230     71     10      1
6:   0   192    784   1312   1140    532    123     12     1
7:   0   448   2144   4368   4872   3164   1162    211    14    1
8:   0  1024   5632  13568  18592  15680   8176   2480   367   16   1
9:   0  2304  14336  39936  65088  67872  46368  20304  5262  655  18  1

G. C. Greubel, Rows n=0..100 of triangle, flattened
James Kaiser, Jessica S. Purcell, Clint Rollins, Volumes of chain links, arXiv:1107.2865 [math.GT], 2011.
Louis H. Kauffman, Introductory Lectures on Knot Theory, Selected Lectures Presented at the Advanced School and Conference on Knot Theory and Its Applications to Physics and Biology, World Scientific, 2012.
Louis H. Kauffman and Vassily O. Manturov, New Ideas in Low Dimensional Topology, World Scientific, 2015.
Carolina Medina, Jorge Ramírez-Alfonsín and Gelasio Salazar, On the number of unknot diagrams, arXiv:1710.06470 [math.CO], 2017.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.
Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.

Row sums: A000302 (powers of 4).

Cf. A299989, A300192, A300453, A300454, A316659, A316989.

With[{nmax = 15}, CoefficientList[CoefficientList[Series[(x^2 + y*x/(1 - y*(2*x + 2)))/(1 - y*(x + 2)), {x, 0, nmax + 2}, {y, 0, nmax}], y], x]] // Flatten (* G. C. Greubel, Oct 18 2018 *)
Table[SeriesCoefficient[x^2*(x+2)^n + x*Sum[(x+2)^(n-j-1)*(2*x+2)^j, {j, 0, n-1}], {x, 0, k}], {n, 0, 10}, {k, 0, n+2}]//Flatten  (* Michael De Vlieger, Oct 20 2018 *)

T(n, k) := ratcoef((2*x + 2)^n + (x^2 - 1)*(x + 2)^n, x, k)$
create_list(T(n, k), n, 0, 10, k, 0, n + 2);

{T(n,k) = if(k==0, 0, if(k==1, n*2^(n-1), if(k==n+2, 1, T(n-1, k-1) + 2*T(n-1,k) + 2^(n-1)*binomial(n-1,k-1) )))};
for(n=0,10, for(k=0,n+2, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 20 2018

T(n,0) = 0, T(n,1) = n*2^(n-1), T(0,2) = 1 and T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + A038208(n-1,k-1).
T(n,1) = A001787(n).
T(n,n) = A295077(n).
T(n,n+1) = A005843(n).
G.f.: (x^2 + y*x/(1 - y*(2*x + 2)))/(1 - y*(x + 2)).

New name by Franck Maminirina Ramaharo, Oct 17 2018

A130123 Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).

Original entry on oeis.org

1, 0, 2, 0, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 128, 0, 0, 0, 0, 0, 0, 0, 0, 256, 0, 0, 0, 0, 0, 0, 0, 0, 0, 512, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2048, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4096

Offset: 0

Gary W. Adamson, May 11 2007

A 2^n transform matrix.
A130123 * A007318 = A038208. A007318 * A130123 = A013609. A130124 = A130123 * A002260. A130125 = A128174 * A130123.
Triangle T(n,k), 0 <= k <= n, given by [0,0,0,0,0,0,...] DELTA [2,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, May 26 2007
Also the Bell transform of A000038. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016
T is the convolution triangle of the characteristic function of 2 (see A357368). - Peter Luschny, Oct 19 2022

			First few terms of the triangle:
  1;
  0, 2;
  0, 0, 4;
  0, 0, 0, 8;
  0, 0, 0, 0, 16;
  0, 0, 0, 0,  0, 32; ...

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

Cf. A130124, A130125.

[[k eq n select 2^n else 0: k in [0..n]]: n in [0..14]]; // G. C. Greubel, Jun 05 2019

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n=0,2,0), 9); # Peter Luschny, Jan 27 2016
# Uses function PMatrix from A357368.
PMatrix(10, n -> ifelse(n=1, 2, 0)); # Peter Luschny, Oct 19 2022

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[# == 0, 2, 0]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
Table[If[k==n, 2^n, 0], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 05 2019 *)

{T(n,k) = if(k==n, 2^n, 0)}; \\ G. C. Greubel, Jun 05 2019

def T(n, k):
    if (k==n): return 2^n
    else: return 0
[[T(n, k) for k in (0..n)] for n in (0..14)] # G. C. Greubel, Jun 05 2019

G.f.: 1/(1-2*x*y). - R. J. Mathar, Aug 11 2015

A154690 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*binomial(n,k), 0 <= k <= n.

Original entry on oeis.org

2, 3, 3, 5, 8, 5, 9, 18, 18, 9, 17, 40, 48, 40, 17, 33, 90, 120, 120, 90, 33, 65, 204, 300, 320, 300, 204, 65, 129, 462, 756, 840, 840, 756, 462, 129, 257, 1040, 1904, 2240, 2240, 2240, 1904, 1040, 257, 513, 2322, 4752, 6048, 6048, 6048, 6048, 4752, 2322, 513

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jan 14 2009

From G. C. Greubel, Jan 18 2025: (Start)
A more general triangle of coefficients may be defined by T(n, k, p, q) = (p^(n-k)*q^k + p^k*q^(n-k))*A007318(n, k). When (p, q) = (2, 1) this sequence is obtained.
Some related triangles are:
  (p, q) = (1, 1) : 2*A007318(n,k).
  (p, q) = (2, 2) : 2*A038208(n,k).
  (p, q) = (3, 2) : A154692(n,k).
  (p, q) = (3, 3) : 2*A038221(n,k). (End)

			Triangle begins as:
     2;
     3,    3;
     5,    8,     5;
     9,   18,    18,     9;
    17,   40,    48,    40,    17;
    33,   90,   120,   120,    90,    33;
    65,  204,   300,   320,   300,   204,    65;
   129,  462,   756,   840,   840,   756,   462,   129;
   257, 1040,  1904,  2240,  2240,  2240,  1904,  1040,   257;
   513, 2322,  4752,  6048,  6048,  6048,  6048,  4752,  2322,  513;
  1025, 5140, 11700, 16320, 16800, 16128, 16800, 16320, 11700, 5140, 1025;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Lakhtakia, R. Messier, V. K. Varadan, and V. V. Varadan, Use of combinatorial algebra for diffusion on fractals, Physical Review A, volume 34, Number 3 (1986) p. 2502, Fig. 3.

Cf. A000129, A001045, A025192, A038208, A038221, A069720, A107920, A154692.
Cf. A215149.
Sums include: A008776 (row), A010673 (alternating sign row).
Columns k: A000051 (k=0).
Main diagonal: A059304.

A154690:= func< n,k | (2^(n-k)+2^k)*Binomial(n,k) >;
[A154690(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 18 2025

A154690 := proc(n,m) binomial(n,m)*(2^(n-m)+2^m) ; end proc: # R. J. Mathar, Jan 13 2011

T[n_, m_]:= (2^(n-m) + 2^m)*Binomial[n,m];
Table[T[n,m], {n,0,12}, {m,0,n}]//Flatten

from sage.all import *
def A154690(n,k): return (pow(2,n-k)+pow(2,k))*binomial(n,k)
print(flatten([[A154690(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025

T(n, k) = (2^(n-k) + 2^k)*A007318(n, k).
Sum_{k=0..n} T(n, k) = A008776(n) = A025192(n+1).
From G. C. Greubel, Jan 18 2025: (Start)
T(n, n-k) = T(n, k) (symmetry).
T(n, 1) = n + A215149(n), n >= 1.
T(2*n-1, n) = 3*A069720(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A010673(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000129(n+1) + A001045(n+1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = n+1 + A107920(n+1). (End)

A099040 Riordan array (1, 2+2x).

Original entry on oeis.org

1, 0, 2, 0, 2, 4, 0, 0, 8, 8, 0, 0, 4, 24, 16, 0, 0, 0, 24, 64, 32, 0, 0, 0, 8, 96, 160, 64, 0, 0, 0, 0, 64, 320, 384, 128, 0, 0, 0, 0, 16, 320, 960, 896, 256, 0, 0, 0, 0, 0, 160, 1280, 2688, 2048, 512, 0, 0, 0, 0, 0, 32, 960, 4480, 7168, 4608, 1024, 0, 0, 0, 0, 0, 0, 384, 4480, 14336, 18432, 10240, 2048

Offset: 0

Paul Barry, Sep 23 2004

Row sums give A002605. Diagonal sums give A052907.
The Riordan array (1,s+t*x) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).
T(n,k) is the number of compositions of n into two types of parts of size 1 and 2 that have exactly k parts. - Geoffrey Critzer, Aug 18 2012.
Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, ...] DELTA [2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 22 2020

			Rows begin {1}, {0,2}, {0,2,4}, {0,0,8,8}, {0,0,4,24,16}, {0,0,0,24,64,32},...
T(3,2)=8 because we have: 1+2,1+2',1'+2,1'+2',2+1,2+1',2'+1,2'+1' where a part of the second type is designated by '. - _Geoffrey Critzer_, Aug 18 2012

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A002605, A026729, A038208, A052907.

nn = 8; CoefficientList[Series[1/(1 - 2 y x - 2 y x^2), {x, 0, nn}], {x, y}] // Grid  (* Geoffrey Critzer, Aug 18 2012 *)

Number triangle T(n, k) = 2^k*binomial(k, n-k).
Columns have g.f. (2x+2x^2)^k.
T(n,k) = A026729(n,k)*2^k. - Philippe Deléham, Jul 28 2006
O.g.f.: 1/(1-2*y*x-2*y*x^2). - Geoffrey Critzer, Aug 18 2012.

A308737 Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is the 1-tiered binomial coefficient.

Original entry on oeis.org

1, 1, 3, 1, 8, 7, 1, 17, 31, 15, 1, 34, 96, 94, 31, 1, 67, 258, 382, 253, 63, 1, 132, 645, 1280, 1275, 636, 127, 1, 261, 1545, 3845, 5115, 3831, 1531, 255, 1, 518, 3598, 10766, 17920, 17906, 10738, 3578, 511, 1, 1031, 8212, 28700, 57358, 71666, 57316, 28652, 8185, 1023

Offset: 0

Michel Marcus, Jun 21 2019

			From _Petros Hadjicostas_, Jul 07 2020: (Start)
Square array for (N,M)_1 of 1-tiered binomial coefficients (N, M >= 0):
  1/2,   3/4,    7/8,     15/16,   31/32,     63/64,   127/128, ...
  1/4,    1,    31/16,    47/16,  253/64,    159/32,  1531/256, ...
  1/8,  17/16,    3,     191/32, 1275/128,  3831/256,  5369/256, ...
  1/16, 17/16, 129/32,     10,   5115/256,  8953/256, 14329/256, ...
  1/32, 67/64, 645/128, 3845/256,   35,    35833/512, 129003/1024, ...
  ... (End)
Triangle (n-k,k)_1 of 1-tiered binomial coefficients (n >= 0 and k = 0..n):
  1/2,
  1/4,    3/4,
  1/8,     1,    7/8,
  1/16,  17/16, 31/16, 15/16,
  1/32,  17/16,   3,   47/16, 31/32,
  ...
Scaled triangle T(n,k) after multiplying each row by 2^(n+1):
  1,
  1,  3,
  1,  8,  7,
  1, 17, 31, 15,
  1, 34, 96, 94, 31,
  ...

Michael E. Hoffman, (Poly)logarithmic Integrals and Multiple Zeta Values, Number Theory Talk, Max-Planck-Institut für Mathematik, Bonn, 20 June 2018. See Slide 29.
Michael E. Hoffman and Markus Kuba, Logarithmic integrals, zeta values, and tiered binomial coefficients, arXiv:1906.08347 [math.CO], 2019-2020. See Example 4 on pp. 12-13.

Cf. A007318 (Pascal triangle: 0-tiered binomial coefficient), A038208, A145071 (column k = 1).

rows = 10;
cc = CoefficientList[# + O[y]^rows, y]& /@ CoefficientList[(1-x)/((1-x-y)* (2-x-y)) + O[x]^rows, x];
T[n_, m_, 1] := cc[[n-m+1, m+1]];
Table[2^(n+1) Table[T[n, m, 1], {m, 0, n}], {n, 0, rows-1}] (* Jean-François Alcover, Jun 21 2019 *)

T(n,m) = if ((n==0) && (m==0), 1/2, binomial(n+m-1, m-1) - (binomial(n+m,n)/2 - binomial(n+m-1,n-1))/2^(n+m));
TT(n, k) = T(n-k, k);
tabls(nn) = for (n=0, nn, for (k=0, n, print1(2^(n+1)*TT(n, k), ", ")));

(N,M)1 = ((N-1,M)_1 + (N,M-1)_1 + (N,M-1)_0)/2 for N, M >= 0 and N + M > 0 with initial conditions: (N,-1)_i = 0 = (-1,M)_i for i = 0, 1; (0,0)_1 = 1/2; and (N,M)_0 = binomial(N+M, M) for N, M >= 0. (This is the recurrence for the square array presentation of the regular triangle.) [Edited by _Petros Hadjicostas, Jul 06 2020]
(N,M)1 = binomial(N+M-1, M-1) - (binomial(N+M, N)/2 - binomial(N+M-1, N-1))/2^(N+M) for N,M >= 0 and N + M > 0 with (0,0)_1 = 1/2. - _Petros Hadjicostas, Jul 06 2020
G.f. for (N,M)1: (1-x)/((1-x-y)*(2-x-y)). - _Jean-François Alcover, Jun 21 2019
Scaled coefficients satisfy T(n,0) = 1 for n >= 0 and T(n,k) = T(n-1,k) + T(n-1,k-1) + 2^n*C(n-1,k-1) for n >= k+1 >= 1. - Charlie Neder, Jun 21 2019 [Corrected by Petros Hadjicostas, Jul 06 2020]
From Petros Hadjicostas, Jul 07 2020: (Start)
(N,M)_1 + (M,N)_1 = (N,M)_0 = binomial(N+M, N) for N, M >= 0.
(n-k,k)_1 + (k, n-k)_1 = binomial(n,k) for n >= k >= 0.
T(n,k) + T(n,n-k) = 2^(n+1)*binomial(n,k) = 2*A038208(n,k) for n >= k >= 0.
T(n,k) = 2^(n + 1)*binomial(n-1, k-1) + 2*binomial(n-1,k) - binomial(n,k) for n >= k >= 0 and (n,k) <> (0,0) with T(0,0) = 1.
G.f. for T(n,k): (1 - 2*x)/((1 - 2*x*(1 + y))*(1 - x*(1 + y))). (End)

Name edited by Petros Hadjicostas, Jul 07 2020

cp's OEIS Frontend

A257618 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.

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A300184 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (2*x + 2)^n + (x^2 - 1)*(x + 2)^n.

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A130123 Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).

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A154690 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*binomial(n,k), 0 <= k <= n.

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A099040 Riordan array (1, 2+2x).

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A308737 Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is the 1-tiered binomial coefficient.

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A257618 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = 8*x + 2.

A300184 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (2x + 2)^n + (x^2 - 1)(x + 2)^n.