A054879 -id:A054879 - cp's OEIS frontend

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.

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

A328778 Number of indecomposable closed walks of length 2n along the edges of a cube based at a vertex.

Original entry on oeis.org

1, 3, 12, 84, 588, 4116, 28812, 201684, 1411788, 9882516, 69177612, 484243284, 3389702988, 23727920916, 166095446412, 1162668124884, 8138676874188, 56970738119316, 398795166835212, 2791566167846484, 19540963174925388

Offset: 0

Geoffrey Critzer, Oct 27 2019

An indecomposable closed walk returns to its starting vertex exactly once (on the final step).

For n > 1, a(n) is the number of 4-colorings of the grid graph P_2 X P_(n-1). More generally, for q > 1, the number of q-colorings of the grid graph P_2 X P_n is given by q*(q - 1)*((q - 1)*(q - 2) + 1)^(n - 1). - Sela Fried, Sep 25 2023

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7).

Cf. A054879, A025192.

nn = 40; list = Range[0, nn]! CoefficientList[Series[ Cosh[x]^3, {x, 0, nn}], x]; a = Sum[list[[i]] x^(i - 1), {i, 1, nn + 1}]; Select[CoefficientList[Series[ 2 - 1/a, {x, 0, nn}], x], # > 0 &]

Vec((1 - 4*x - 9*x^2) / (1 - 7*x) + O(x^25)) \\ Colin Barker, Oct 28 2019

G.f.: 2 - 1/f(x) where f(x) is the g.f. for A054879.
From Colin Barker, Oct 27 2019: (Start)
G.f.: (1 - 4*x - 9*x^2) / (1 - 7*x).
a(n) = 7*a(n-1) for n>2.
a(n) = 12*7^(n - 2) for n>1.
(End)
E.g.f.: (1/49)*(37 + 12*exp(7*x) + 63*x). - Stefano Spezia, Oct 27 2019

A158303 Triangle read by rows, A007318 * (A158300 * 0^(n-k)).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 1, 6, 6, 8, 1, 8, 12, 32, 8, 1, 10, 20, 80, 40, 32, 1, 12, 30, 160, 120, 192, 32, 1, 14, 42, 280, 280, 672, 224, 128, 1, 16, 56, 448, 560, 1792, 896, 1024, 128, 1, 18, 72, 672, 1008, 4032, 2688, 4608, 1152, 512

Offset: 0

Gary W. Adamson, Mar 15 2009

			First few rows of the triangle =
1;
1, 2;
1, 4, 2;
1, 6, 6, 8;
1, 8, 12, 32, 8;
1, 10, 20, 80, 40, 32;
1, 12, 30, 160, 120, 192, 32;
1, 14, 42, 280, 280, 672, 224, 128;
1, 16, 56, 448, 560, 1792, 896, 1024, 128;
1, 18, 72, 672, 1008, 4032, 2688, 4608, 1152, 512;
1, 20, 90, 960, 1680, 8064, 6720, 15360, 5760, 5120, 512;
...

Cf. A158300, A122983 (row sums), A054879, A066443

Triangle read by rows, A007318 * (A158300 * 0^(n-k)). Equals binomial transform of an infinite lower triangular matrix with A158300: (1, 2, 2, 8, 8, 32, 32,...) as the main diagonal and the rest zeros.

A328821 Triangular array read by rows. Let P be the poset of all even sized subsets of [2n] ordered by inclusion. T(n,k) is the number of intervals in P with length k, 0<=k<=n, n>=0.

Original entry on oeis.org

1, 2, 1, 8, 12, 1, 32, 120, 30, 1, 128, 896, 560, 56, 1, 512, 5760, 6720, 1680, 90, 1, 2048, 33792, 63360, 29568, 3960, 132, 1, 8192, 186368, 512512, 384384, 96096, 8008, 182, 1, 32768, 983040, 3727360, 4100096, 1647360, 256256, 14560, 240, 1

Offset: 0

Geoffrey Critzer, Jun 07 2020

			1,
2,   1,
8,   12,   1,
32,  120,  30,   1,
128, 896,  560,  56,   1,
512, 5760, 6720, 1680, 90, 1

Cf. A054879 (row sums), A081294 (column k=0).

nn = 8; ev[x_] := Sum[x^n/((2 n)!/2^n), {n, 0, nn}];
Map[Select[#, # > 0 &] &, Table[(2 n)!/2^n, {n, 0, nn}] CoefficientList[Series[ev[x]^2 ev[y x], {x, 0, nn}], {x, y}]] // Flatten

Let E(x) = Sum_{n>=0} x^n/((2n)!/2^n). Then Sum_{n>=0} Sum{k=0..n} T(n,k) y^k*x^n/((2n)!/2^n) = E(y*x) * E(x)^2.

A336667 Triangular array read by rows. T(n,k) is the number of closed walks of length 2n along the edges of a cube based at vertex v that return to v exactly k times, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 3, 0, 12, 9, 0, 84, 72, 27, 0, 588, 648, 324, 81, 0, 4116, 5544, 3564, 1296, 243, 0, 28812, 45864, 35748, 16848, 4860, 729, 0, 201684, 370440, 337932, 193104, 72900, 17496, 2187

Offset: 0

Geoffrey Critzer, Jul 29 2020

			Triangle T(n,k) begins:
  1;
  0,   3;
  0,  12,   9;
  0,  84,  72,  27;
  0, 588, 648, 324, 81;
  ...

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; page 340.

Cf. A054879 (row sums), A328778 (column k=1).

Table[nn = n; CoefficientList[Series[(1 - 7 z^2)/(1 - (7 + 3 u) z^2 + 9 u z^4), {z, 0, nn}], {z,u}][[-1]], {n, 0, 15, 2}] // Grid

O.g.f.: (1 - 7*x^2)/(1 - 7*x^2 - 3*y*x^2 + 9*y*x^4).

A377946 Cogrowth sequence of the 16-element group C2^2:C4 = .

Original entry on oeis.org

1, 1, 2, 10, 40, 136, 512, 2080, 8320, 32896, 131072, 524800, 2099200, 8390656, 33554432, 134225920, 536903680, 2147516416, 8589934592, 34359869440, 137439477760, 549756338176, 2199023255552, 8796095119360, 35184380477440, 140737496743936, 562949953421312

Offset: 0

Sean A. Irvine, Nov 11 2024

Gives the even terms, all the odd terms are 0.

Also called K8:C2. Gap identifier 16, 3.

Cf. A377855 (C4:C4), A054879 (C2^3), A377843 (C4 X C2^2), A377943 (C4 o D4).

G.f.: (12*x^4-14*x^3+8*x^2-5*x+1) / ((4*x-1) * (2*x-1) * (4*x^2+1)).

cp's OEIS Frontend

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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A328778 Number of indecomposable closed walks of length 2n along the edges of a cube based at a vertex.

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A158303 Triangle read by rows, A007318 * (A158300 * 0^(n-k)).

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A328821 Triangular array read by rows. Let P be the poset of all even sized subsets of [2n] ordered by inclusion. T(n,k) is the number of intervals in P with length k, 0<=k<=n, n>=0.

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A336667 Triangular array read by rows. T(n,k) is the number of closed walks of length 2n along the edges of a cube based at vertex v that return to v exactly k times, n>=0, 0<=k<=n.

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Formula

A377946 Cogrowth sequence of the 16-element group C2^2:C4 = .

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