A121822 -id:A121822 - cp's OEIS frontend

A081294 Expansion of (1-2x)/(1-4x).

1, 2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728, 536870912, 2147483648, 8589934592, 34359738368, 137438953472, 549755813888, 2199023255552, 8796093022208, 35184372088832

Offset: 0

Paul Barry, Mar 17 2003

Binomial transform of A046717. Second binomial transform of A000302 (with interpolated zeros). Partial sums are A007583.
Counts closed walks of length 2n at a vertex of the cyclic graph on 4 nodes C_4. With interpolated zeros, counts closed walks of length n at a vertex of the cyclic graph on 4 nodes C_4. - Paul Barry, Mar 10 2004
In general, Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, j)*r^j has expansion (1 - (r+1)*x)/(1 - (r+3)*x - (r-1)*(r+3)*x^2 + (r-1)^3*x^3). - Paul Barry, Jun 04 2005 [corrected by Jason Yuen, Jan 20 2025]
a(n) is the number of binary strings of length 2n with an even number of 0's (and hence an even number of 1's). - Toby Gottfried, Mar 22 2010
Number of compositions of n where there are 2 sorts of part 1, 4 sorts of part 2, 8 sorts of part 3, ..., 2^k sorts of part k. - Joerg Arndt, Aug 04 2014
a(n) is also the number of permutations simultaneously avoiding 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl Aug 07 2014
INVERT transform of powers of 2 (A000079). - Alois P. Heinz, Feb 11 2021
a(n) is the number of elements in an n-interval of the binomial poset of even-sized subsets of positive integers, cf. Stanley reference and second formula by Paul Barry.  Each multichain 0 = x_0 <= x_1 <= x_2 = 1 in such an n-interval corresponds to a closed walk described above by Paul Barry.  More generally, each multichain 0 = x_0 <= x_1 <= ... <= x_k = 1 corresponds to a closed walk of length 2n on the k-dimensional hypercube, cf. A054879, A092812, A121822. - Geoffrey Critzer, Apr 21 2023

			G.f. = 1 + 2*x + 8*x^2 + 32*x^3 + 128*x^4 + 512*x^5 + 2048*x^6 + 8192*x^7 + ...

Richard P. Stanley, Enumerative Combinatorics, Vol 1, second edition, Example 3.18.3-f, page 323.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
M. Paukner, L. Pepin, M. Riehl, and J. Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015-2016.
Index to divisibility sequences.
Index entries for linear recurrences with constant coefficients, signature (4).

Row sums of triangle A136158.

Cf. A000079, A081295, A009117, A016742, A054879, A092812, A121822. Essentially the same as A004171.

[(4^n+0^n)/2: n in [0..30]]; // Vincenzo Librandi, Jul 26 2011

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1-2*x)/(1-4*x))); // Marius A. Burtea, Jan 20 2020

a:= n-> 2^max(0, (2*n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 20 2017

CoefficientList[Series[(1-2x)/(1-4x),{x,0,40}],x] (* or *)
Join[{1}, NestList[4 # &, 2, 40]] (* Harvey P. Dale, Apr 22 2011 *)

a(n)=1<Charles R Greathouse IV, Jul 25 2011

x='x+O('x^100); Vec((1-2*x)/(1-4*x)) \\ Altug Alkan, Dec 21 2015

G.f.: (1-2*x)/(1-4*x).
a(n) = 4*a(n-1) n > 1, with a(0)=1, a(1)=2.
a(n) = (4^n+0^n)/2 (i.e., 1 followed by 4^n/2, n > 0).
E.g.f.: exp(2*x)*cosh(2*x) = (exp(4*x)+exp(0))/2. - Paul Barry, May 10 2003
a(n) = Sum_{k=0..n} C(2*n, 2*k). - Paul Barry, May 20 2003
a(n) = A001045(2*n+1) - A001045(2*n-1) + 0^n/2. - Paul Barry, Mar 10 2004
a(n) = 2^n*A011782(n); a(n) = gcd(A011782(2n), A011782(2n+1)). - Paul Barry, Jan 12 2005
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, j). - Paul Barry, Jun 04 2005
a(n) = Sum_{k=0..n} A038763(n,k). - Philippe Deléham, Sep 22 2006
a(n) = Integral_{x=0..4} p(n,x)^2/(Pi*sqrt(x(4-x))) dx, where p(n,x) is the sequence of orthogonal polynomials defined by C(2*n,n): p(n,x) = (2*x-4)*p(n-1,x) - 4*p(n-2,x), with p(0,x)=1, p(1,x)=-2+x. - Paul Barry, Mar 01 2007
a(n) = ((2+sqrt(4))^n + (2-sqrt(4))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Nov 22 2008
a(n) = A000079(n) * A011782(n). - Philippe Deléham, Dec 01 2008
a(n) = A004171(n-1) = A028403(n) - A000079(n) for n >= 1. - Jaroslav Krizek, Jul 27 2009
a(n) = Sum_{k=0..n} A201730(n,k)*3^k. - Philippe Deléham, Dec 06 2011
a(n) = Sum_{k=0..n} A134309(n,k)*2^k = Sum_{k=0..n} A055372(n,k). - Philippe Deléham, Feb 04 2012
G.f.: Q(0), where Q(k) = 1 - 2*x/(1 - 2/(2 - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 29 2013
E.g.f.: 1/2 + exp(4*x)/2 = (Q(0)+1)/2, where Q(k) = 1 + 4*x/(2*k+1 - 2*x*(2*k+1)/(2*x + (k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 29 2013
a(n) = ceiling( 2^(2n-1) ). - Wesley Ivan Hurt, Jun 30 2013
G.f.: 1 + 2*x/(1 + x)*( 1 + 5*x/(1 + 4*x)*( 1 + 8*x/(1 + 7*x)*( 1 + 11*x/(1 + 10*x)*( 1 + ... )))). - Peter Bala, May 27 2017
Sum_{n>=0} 1/a(n) = 5/3. - Amiram Eldar, Aug 18 2022
Sum_{n>=0} a(n)*x^n/A000680(n) = E(x)^2 where E(x) = Sum_{n>=0} x^n/A000680(n). - Geoffrey Critzer, Apr 21 2023

A054879 Closed walks of length 2n along the edges of a cube based at a vertex.

Original entry on oeis.org

1, 3, 21, 183, 1641, 14763, 132861, 1195743, 10761681, 96855123, 871696101, 7845264903, 70607384121, 635466457083, 5719198113741, 51472783023663, 463255047212961, 4169295424916643, 37523658824249781, 337712929418248023, 3039416364764232201, 27354747282878089803, 246192725545902808221

Offset: 0

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

a(n) is the number of words of length 2n on alphabet {0,1,2} with an even number (possibly zero) of each letter. - Geoffrey Critzer, Dec 20 2012

Equivalently, the cogrowth sequence of the 8-element group C2^3. - Sean A. Irvine, Nov 04 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. Benkart and D. Moon, A Schur-Weyl Duality Approach to Walking on Cubes, arXiv preprint arXiv:1409.8154 [math.RT], 2014 and Ann. Combin. 20 (3) (2016) 397-417
R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015) # 15.3.2.
G. R. Franssens, On a number pyramid related to the binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
Katarzyna Grygiel, Pawel M. Idziak and Marek Zaionc, How big is BCI fragment of BCK logic, arXiv preprint arXiv:1112.0643 [cs.LO], 2011. [From _N. J. A. Sloane_, Feb 21 2012]
Ji-Hwan Jung, Oriented Riordan graphs and their fractal property, arXiv:2009.01677 [math.CO], 2020.
R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 5.
L. Reyzin, Number of closed walks on an n-cube, Mathoverflow.
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A081294, A092812, A121822, A122983.

[(3^(2*n)+3)/4: n in [0..25]]; // Vincenzo Librandi, Jun 30 2011

nn = 40; Select[Range[0, nn]! CoefficientList[Series[Cosh[x]^3, {x, 0, nn}], x], # > 0 &]  (* Geoffrey Critzer, Dec 20 2012 *)
Table[(3^(2n)+3)/4,{n,0,30}] (* or *) LinearRecurrence[{10,-9},{1,3},30] (* Harvey P. Dale, Mar 17 2023 *)

a(n) = (3^(2*n)+3)/4.
G.f.: 1/4*1/(1-9*x)+3/4*1/(1-x).
a(n) = Sum_{k=0..n} 3^k*4^(n-k)*A121314(n,k). - Philippe Deléham, Aug 26 2006
E.g.f.: cosh^3(x). O.g.f.: 1/(1-3*1*x/(1-2*2*x/(1-1*3*x))) (continued fraction). - Peter Bala, Nov 13 2006
(-1)^n*a(n) = Sum_{k=0..n} A086872(n,k)*(-4)^(n-k). - Philippe Deléham, Aug 17 2007
a(n) = (1/2^3)*Sum_{j = 0..3} binomial(3,j)*(3 - 2*j)^(2*n). See Reyzin link. - Peter Bala, Jun 03 2019
a(n) = 9*a(n-1) - 6. - Klaus Purath, Mar 13 2021

A092812 Number of closed walks of length 2*n on the 4-cube.

Original entry on oeis.org

1, 4, 40, 544, 8320, 131584, 2099200, 33562624, 536903680, 8590065664, 137439477760, 2199025352704, 35184380477440, 562949986975744, 9007199388958720, 144115188612726784, 2305843011361177600

Offset: 0

Paul Barry, Mar 11 2004

With interpolated zeros this has a(n) = (6*0^n + 4^n + (-4)^n + 4*2^n + 4*(-2)^n)/16 and counts closed walks of length n at a vertex of the 4-cube. [Typo corrected by Alexander R. Povolotsky, May 26 2008]

Also, cogrowth sequence of the 16-element group C2^4. - Sean A. Irvine, Nov 10 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..200
G. R. Franssens, On a number pyramid related to the binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
Katarzyna Grygiel, Pawel M. Idziak and Marek Zaionc, How big is BCI fragment of BCK logic, arXiv preprint arXiv:1112.0643 [cs.LO], 2011. [From _N. J. A. Sloane_, Feb 21 2012]
L. Reyzin, Mathoverflow, Number of closed walks on an n-cube.
Index entries for linear recurrences with constant coefficients, signature (20,-64).

Essentially the same as A075878.

Cf. A026244, A081294, A054879, A121822.

[3*0^n/8+16^n/8+4^n/2: n in [0..30]]; // Vincenzo Librandi, May 31 2011

CoefficientList[Series[(1-16x+24x^2)/((1-4x)(1-16x)),{x,0,30}],x] (* or *) Join[{1},LinearRecurrence[{20,-64},{4,40},30]] (* Harvey P. Dale, Aug 23 2011 *)

G.f.: (1-16*x+24*x^2)/((1-4*x)*(1-16*x)).
a(n) = 3*0^n/8 + 16^n/8 + 4^n/2.
From Peter Bala, Nov 13 2006: (Start)
E.g.f.: cosh^4(x).
O.g.f.: 1/(1-4*1*x/(1-3*2*x/(1-2*3*x/(1-1*4*x)))) (continued fraction). (End)
(-1)^n*a(n) = Sum_{k=0..n} A086872(n,k)*(-5)^(n-k). - Philippe Deléham, Aug 17 2007
a(n) = 20*a(n-1) - 64*a(n-2); a(0) = 1, a(1) = 4, a(2) = 40. - Harvey P. Dale, Aug 23 2011
a(n) = 4*A026244(n-1), n > 0. - R. J. Mathar, Oct 24 2014
a(n) = (1/2^4)*Sum_{j = 0..4} binomial(4, j)*(4 - 2*j)^(2*n). See Reyzin link. - Peter Bala, Jun 03 2019

Title improved by Sean A. Irvine at the suggestion of Peter Bala, Jun 04 2019

A086872 Triangle T(n, k) read by rows; given by [1, 2, 3, 4, 5, 6, ..] DELTA [1, 4, 9, 16, 25, 36, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 3, 8, 5, 15, 75, 121, 61, 105, 840, 2478, 3128, 1385, 945, 11025, 51030, 115350, 124921, 50521, 10395, 166320, 1105335, 3859680, 7365633, 7158128, 2702765

Offset: 0

Philippe Deléham, Aug 20 2003, Aug 17 2007

			Triangle begins:
1;
1, 1;
3, 8, 5;
15, 75, 121, 61;
105, 840, 2478, 3128, 1385;
945, 11025, 51030, 115350, 124921, 50521;
10395, 166320, 1105335, 3859680, 7365633, 7158128, 2702765 ; ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000182 (row sums), A000364 (first diagonal), A001147 (first column), A084938, A261065 (2nd column).

Sum( k>=0, T(n, k)*(-1)^k ) = 0; if n>0.
Sum( k>=0, T(n, k)*(-1/2)^k ) = (1/2)^n.
Sum_{k, 0<=k<=n}T(n,k)*x^(n-k) = (-1)^n*A121822(n), (-1)^n*A092812(n), (-1)^n*A054879(n), A009117(n), A033999(n), A000007(n), A000364(n), A000182(n+1) for x = -6, -5, -4, -3, -2, -1, 0, 1 respectively .

A286899 Array read by antidiagonals: A(n, L) is the number of closed walks of length 2L along the edges of an n-cube based at a vertex, for n >= 1 and L >= 1.

Original entry on oeis.org

1, 1, 2, 1, 8, 3, 1, 32, 21, 4, 1, 128, 183, 40, 5, 1, 512, 1641, 544, 65, 6, 1, 2048, 14763, 8320, 1205, 96, 7, 1, 8192, 132861, 131584, 26465, 2256, 133, 8, 1, 32768, 1195743, 2099200, 628805, 64896, 3787, 176, 9, 1, 131072, 10761681, 33562624, 15424865

Offset: 1

Melvin Peralta, May 15 2017

			A(2, 2) = 8 because at each vertex of a 2-cube (i.e., a square), there are 8 closed walks of length 2(2) = 4.
A(1, k) = 1 because at the vertex of a 1-cube, there is 1 closed walk of any length 2*k.
Array A(n, L) begins:
   1         1         1         1         1         1 ...
   2         8        32       128       512      2048 ...
   3        21       183      1641     14763    132861 ...
   4        40       544      8320    131584   2099200 ...
   5        65      1205     26465    628805  15424865 ...
   6        96      2256     64896   2086656  71172096 ...
   7       133      3787    134953   5501167 243147373 ...

R. P. Stanley, Algebraic Combinatorics: Walks, Trees, Tableaux, and More, Springer, 2013.

Melvin Peralta, Table of n, a(n) for n = 1..120

Cf. A054879, A092812, A121822.

A286899 := proc(n,L)
    add(binomial(n,i)*(n-2*i)^L, i=0..n) ;
    %/2^n ;
end proc:
for n from 1 to 7 do
    for L from 2 to 12 by 2 do
        printf("%9d ",A286899(n,L)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, May 22 2017

f[n_, l_] := 1/2^n*Sum[Binomial[n, i]*(n - 2 i)^l, {i, 0, n}];
Table[f[n - l + 1, 2 l], {n, 1, 15}, {l, n, 1, -1}] // Flatten

A(n, L) = (1/2^n)*Sum_{i=0..n} binomial(n, i)*(n - 2*i)^(2*L). (Corrected by Peter Luschny, Jul 07 2019.)

A326476 A(n, k) = (m*k)! [x^k] MittagLefflerE(m, x)^n, for m = 2, n >= 0, k >= 0; square array read by descending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 8, 3, 1, 0, 1, 32, 21, 4, 1, 0, 1, 128, 183, 40, 5, 1, 0, 1, 512, 1641, 544, 65, 6, 1, 0, 1, 2048, 14763, 8320, 1205, 96, 7, 1, 0, 1, 8192, 132861, 131584, 26465, 2256, 133, 8, 1, 0, 1, 32768, 1195743, 2099200, 628805, 64896, 3787, 176, 9, 1

Offset: 0

Peter Luschny, Jul 08 2019

			Array starts:
  [0] 1, 0,   0,    0,      0,        0,          0,            0, ... A000007
  [1] 1, 1,   1,    1,      1,        1,          1,            1, ... A000012
  [2] 1, 2,   8,   32,    128,      512,       2048,         8192, ... A081294
  [3] 1, 3,  21,  183,   1641,    14763,     132861,      1195743, ... A054879
  [4] 1, 4,  40,  544,   8320,   131584,    2099200,     33562624, ... A092812
  [5] 1, 5,  65, 1205,  26465,   628805,   15424865,    382964405, ... A121822
  [6] 1, 6,  96, 2256,  64896,  2086656,   71172096,   2499219456, ...
  [7] 1, 7, 133, 3787, 134953,  5501167,  243147373,  11266376947, ...
  [8] 1, 8, 176, 5888, 250496, 12397568,  676591616,  39316226048, ...
  [9] 1, 9, 225, 8649, 427905, 24943689, 1624354785, 114066126729, ...
        A000567,
Seen as a triangle:
  1;
  0, 1;
  0, 1,    1;
  0, 1,    2,      1;
  0, 1,    8,      3,      1;
  0, 1,   32,     21,      4,     1;
  0, 1,  128,    183,     40,     5,    1;
  0, 1,  512,   1641,    544,    65,    6,   1;
  0, 1, 2048,  14763,   8320,  1205,   96,   7, 1;
  0, 1, 8192, 132861, 131584, 26465, 2256, 133, 8, 1;

Rows n=0..5 give A000007, A000012, A081294, A054879, A092812, A121822.
Columns include: A000567.
Main diagonal gives A381459.
Variant: A286899.
Cf. A326474 (m=3, p>=0), A326475 (m=3, p<=0), A326327 (m=2, p<=0), this sequence (m=2, p>=0).

(* The function MLPower is defined in A326327. *)
For[n = 0, n < 8, n++, Print[MLPower[2, n, 8]]]

a(n, k) = (2*k)!*polcoef(cosh(x+x*O(x^(2*k)))^n, 2*k); \\ Seiichi Manyama, May 11 2025

# uses[MLPower from A326327]
for n in (0..6): print(MLPower(2, n, 9))

A(n,k) = (2*k)! * [x^(2*k)] cosh(x)^n. - Seiichi Manyama, May 11 2025

cp's OEIS Frontend

A081294 Expansion of (1-2*x)/(1-4*x).

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A054879 Closed walks of length 2n along the edges of a cube based at a vertex.

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A092812 Number of closed walks of length 2*n on the 4-cube.

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A086872 Triangle T(n, k) read by rows; given by [1, 2, 3, 4, 5, 6, ..] DELTA [1, 4, 9, 16, 25, 36, ...] where DELTA is the operator defined in A084938.

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A286899 Array read by antidiagonals: A(n, L) is the number of closed walks of length 2L along the edges of an n-cube based at a vertex, for n >= 1 and L >= 1.

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A326476 A(n, k) = (m*k)! [x^k] MittagLefflerE(m, x)^n, for m = 2, n >= 0, k >= 0; square array read by descending antidiagonals.

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A081294 Expansion of (1-2x)/(1-4x).