A124929 -id:A124929 - cp's OEIS frontend

A027649 a(n) = 2*(3^n) - 2^n.

1, 4, 14, 46, 146, 454, 1394, 4246, 12866, 38854, 117074, 352246, 1058786, 3180454, 9549554, 28665046, 86027906, 258149254, 774578834, 2323998646, 6972520226, 20918609254, 62757924914, 188277969046, 564842295746, 1694543664454, 5083664547794, 15251060752246

Offset: 0

N. J. A. Sloane

Poly-Bernoulli numbers B_n^(k) with k=-2.
Binomial transform of A007051, if both sequences start at 0. Binomial transform of A000225(n+1). - Paul Barry, Mar 24 2003
Euler expands (1-z)/(1-5z+6z^2) and finds the general term. Section 226 of the Introductio indicates that he could have written down the recursion relation: a(n) = 5 a(n-1)-6 a(n-2). - V. Frederick Rickey (fred-rickey(AT)usma.edu), Feb 10 2006
Let R be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xRy if x is a subset of y or y is a subset of x. Then a(n) = |R|. - Ross La Haye, Dec 22 2006
With regard to the comment by Ross La Haye: For proper subsets see A056182. - For nonempty subsets see A091344. - For nonempty proper subsets see a(n+1) in A260217. - Manfred Boergens, Aug 02 2023
If x, y are two n-bit binary strings then a(n) gives the number of pairs (x,y) such that XOR(x, y) = ABS(x - y). - Ramasamy Chandramouli, Feb 15 2009
Equals row sums of the triangular version of A038573. - Gary W. Adamson, Jun 04 2009
Inverse binomial transform of A085350. - Paul Curtz, Nov 14 2009
Related to the number of even a's in a nontrivial cycle (should one exist) in the 3x+1 Problem, where a <= floor(log_2(2*(3^n) - 2^n)). The value n correlates to the number of odds in such a nontrivial cycle. See page 1288 of Crandall's paper. Also, this relation gives another proof that the number of odds divided by the number of evens in a nontrivial cycle is bounded by log 2 / log 3 (this observation does not resolve the finite cycles conjecture as the value could be arbitrarily close to this bound). However, the same argument gives that log 2 / log 3 is less than or equal to the number of odds divided by the number of evens in a divergent sequence (should one exist), as log 2 / log 3 is the limit value for a cycle of an arbitrarily large length, where the length is given by the value n. - Jeffrey R. Goodwin, Aug 04 2011
Row sums of Riordan triangle A106516. - Wolfdieter Lang, Jan 09 2015
Number of restricted barred preferential arrangements having 3 bars in which the sections are all restricted sections such that (for fixed sections i and j) section i or section j is empty. - Sithembele Nkonkobe, Oct 12 2015
This is also row 2 of A281891: for n >= 1, when consecutive positive integers are written as a product of primes in nondecreasing order, a factor of 2 or 3 occurs in n-th position a(n) times out of every 6^n. - Peter Munn, May 18 2017
Also row sums of A124929. - Omar E. Pol, Jun 15 2017
This is the sum of A318921(n) for n in the range 2^(k+1) to 2^(k+2)-1. See A318921 for proof. - N. J. A. Sloane, Sep 25 2018
a(n) is also the number of acyclic orientations of the complete bipartite graph K_{2,n}. - Vincent Pilaud, Sep 15 2020
a(n-1) is also the number of n-digit numbers whose largest decimal digit is 2. - Stefano Spezia, Nov 15 2023

Leonhard Euler, Introductio in analysin infinitorum (1748), section 216.

T. D. Noe, Table of n, a(n) for n = 0..200
Taylor Brysiewicz, Holger Eble, and Lukas Kühne, Enumerating chambers of hyperplane arrangements with symmetry, arXiv:2105.14542 [math.CO], 2021.
R. E. Crandall, On the 3x+1 problem, Math. Comp., 32 (1978) 1281-1292.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
John Elias, Illustration of initial terms: Reflected Sierpinski triangle
K. Imatomi, M. Kaneko, and E. Takeda, Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values, J. Int. Seq. 17 (2014) # 14.4.5
Ken Kamano, Sums of Products of Poly-Bernoulli Numbers of Negative Index, Journal of Integer Sequences, Vol. 15 (2012), #12.1.3.
K. Kamano, Sums of Products of Bernoulli Numbers, Including Poly-Bernoulli Numbers, J. Int. Seq. 13 (2010), 10.5.2.
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Takao Komatsu, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
S. Nkonkobe and V. Murali, On some properties and relations between restricted barred preferential arrangements, multi-poly-bernoulli numbers and related numbers, arXiv:1509.07352 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Sierpinski Sieve
Wikipedia, Sierpinski triangle
Index entries for sequences related to Bernoulli numbers.
Index entries for linear recurrences with constant coefficients, signature (5,-6).

Row n = 2 of array A099594.
Also occurs as a row, column, diagonal or as row sums in A038573, A085870, A090888, A106516, A217764, A281891.
Cf. A000079, A000225, A001047, A007051, A053581, A056182, A085350, A091344, A124929, A166060, A260217, A318921.

a027649 n = a027649_list !! n
a027649_list = map fst $ iterate (\(u, v) -> (3 * u + v, 2 * v)) (1, 1)
-- Reinhard Zumkeller, Jun 09 2013

[2*(3^n)-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

a(n, k):= (-1)^n*sum( (-1)^'m'*'m'!*Stirling2(n,'m')/('m'+1)^k,'m'=0..n);
seq(a(n, -2), n=0..30);

Table[2(3^n)-2^n,{n,0,30}] (* or *) LinearRecurrence[ {5,-6},{1,4},31]  (* Harvey P. Dale, Apr 22 2011 *)

a(n)=2*(3^n)-2^n \\ Charles R Greathouse IV, Jul 16 2011

Vec((1-x)/((1-2*x)*(1-3*x)) + O(x^50)) \\ Altug Alkan, Oct 12 2015

[2*(3^n - 2^(n-1)) for n in (0..30)] # G. C. Greubel, Aug 01 2022

G.f.: (1-x)/((1-2*x)*(1-3*x)).
a(n) = 3*a(n-1) + 2^(n-1), with a(0) = 1.
a(n) = Sum_{k=0..n} binomial(n, k)*(2^(k+1) - 1). - Paul Barry, Mar 24 2003
Partial sums of A053581. - Paul Barry, Jun 26 2003
Main diagonal of array (A085870) defined by T(i, 1) = 2^i - 1, T(1, j) = 2^j - 1, T(i, j) = T(i-1, j) + T(i-1, j-1). - Benoit Cloitre, Aug 05 2003
a(n) = A090888(n, 3). - Ross La Haye, Sep 21 2004
a(n) = Sum_{k=0..n} binomial(n+2, k+1)*Sum_{j=0..floor(k/2)} A001045(k-2j). - Paul Barry, Apr 17 2005
a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(n,j)*binomial(j+1,k+1). - Paul Barry, Sep 18 2006
a(n) = A166060(n+1)/6. - Philippe Deléham, Oct 21 2009
a(n) = 5*a(n-1) - 6*a(n-2), a(0)=1, a(1)=4. - Harvey P. Dale, Apr 22 2011
a(n) = A217764(n,2). - Ross La Haye, Mar 27 2013
For n>0, a(n) = 3 * a(n-1) + 2^(n-1) = 2 * (a(n-1) + 3^(n-1)). - J. Conrad, Oct 29 2015
for n>0, a(n) = 2 * (1 + 2^(n-2) + Sum_{x=1..n-2} Sum_{k=0..x-1} (binomial(x-1,k)*(2^(k+1) + 2^(n-x+k)))). - J. Conrad, Dec 10 2015
E.g.f.: exp(2*x)*(2*exp(x) - 1). - Stefano Spezia, May 18 2024

Better formulas from David W. Wilson and Michael Somos
Incorrect formula removed by Charles R Greathouse IV, Mar 18 2010
Duplications (due to corrections to A numbers) removed by Peter Munn, Jun 15 2017

A134346 Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).

Original entry on oeis.org

1, 3, 3, 7, 14, 7, 15, 45, 45, 15, 31, 124, 186, 124, 31, 63, 315, 630, 630, 315, 63, 127, 762, 1905, 2540, 1905, 762, 127, 255, 1785, 5355, 8925, 8925, 5355, 1785, 255, 511, 4088, 14308, 28616, 35770, 28616, 14308, 4088, 511

Offset: 0

Gary W. Adamson, Oct 21 2007

Inverse binomial transform: A134347.
From Wolfdieter Lang, Jul 27 2022: (Start)
Also the triangle t with offset 1 and elements t(n, m) = T(n-1, m-1) read by rows, giving in row n >= 1 the sums of the entries of A356028 of like m.
Also triangle t with offset 1 read by rows, giving in row n >= 1 the sum of the numbers from 1, 2, ..., 2^n - 1 with binary weight m, for m = 1, 2, ..., n. [Observation by Kevin Ryde.] (End)
T(n,k) is the sum of the entries in the (k+2)-th column of the Christmas tree pattern (A367562) of order n+1. - Paolo Xausa, Dec 20 2023

			First few rows of the triangle:
n\k    0    1     2     3      4      5     6     7    8   9 ...
0:     1
1:     3    3
2:     7   14     7
3:    15   45    45    15
4:    31  124   186   124     31
5:    63  315   630   630    315     63
6:   127  762  1905  2540   1905    762   127
7:   255 1785  5355  8925   8925   5355  1785   255
8:   511 4088 14308 28616  35770  28616 14308  4088  511
9:  1023 9207 36828 85932 128898 128898 85932 36828 9207 1023
... reformatted by _Wolfdieter Lang_, Aug 21 2022
----------------------------------------------------------------------------------
T(3, 1) = 12 + 10 + 9 + 6 + 5 + 3 = 45. (From A356028 row n = 4, m = 2.)
Recurrences: T(4, 1) = 45 + 15 + 4*16 = 2*(45 + 15) +4 = 124. - _Wolfdieter Lang_, Jul 27 2022

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).

Cf. A000225, A006516(n+1) (row sums), A124929, A134347, A356028, A356117.

Cf. A367508, A367562.

A134346 := proc(n,k)
    (2^(n+1)-1)*binomial(n,k) ;
end proc:
seq(seq( A134346(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Aug 15 2022
ser := series((1/2 - x)^(k - n - 1) - (1 - x)^(k - n - 1), x, 10):
seq(seq(coeff(ser, x, k), k = 0..n), n = 0..9); # Peter Luschny, Aug 22 2022

A134346[n_,k_]:=(2^(n+1)-1)Binomial[n,k];
Table[A134346[n,k],{n,0,10},{k,0,n}] (* Paolo Xausa, Dec 20 2023 *)

T(n,k) = my(b=binomial(n,k)); b<<(n+1) - b; \\ Kevin Ryde, Aug 15 2022

T(n, m) = A000225(n+1)*A007318(n, m).
From Wolfdieter Lang, Aug 21 2022: (Start)
T(n, k) = 0 for n < k, T(n, 0) = 2^(n+1) - 1, and
T(n, k) = T(n-1, k) + T(n-1, k-1) + binomial(n, k)*2^n, or
T(n, k) = 2*(T(n-1, k) + T(n-1, k-1)) + binomial(n-1, k-1).
(Proof for T(n-1, m-1) = t(n, m), offset 1, by separating in the list of the binary code of the numbers 1, 2, ..., 2^n-1 of length n and weight m the sublists with first entry 1 and 0. The total number of elements of the list for n and m is binomial(n, m).) (End)
T(n, k) = [x^k] ((1/2 - x)^(k - n - 1) - (1 - x)^(k - n - 1)). - Peter Luschny, Aug 22 2022

Name simplified by R. J. Mathar, Aug 15 2022

cp's OEIS Frontend

A134349 A007318^(-1) * (A007318*A124929 + A124929*A007318 - A007318).

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A027649 a(n) = 2*(3^n) - 2^n.

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A134346 Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).

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A134349 A007318^(-1) * (A007318A124929 + A124929A007318 - A007318).