cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Peter Luschny, Jul 08 2019

Keywords

Examples

			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;

Crossrefs

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).

Programs

  • Mathematica
    (* The function MLPower is defined in A326327. *)
For[n = 0, n < 8, n++, Print[MLPower[2, n, 8]]]
  • PARI
    a(n, k) = (2*k)!*polcoef(cosh(x+x*O(x^(2*k)))^n, 2*k); \\ Seiichi Manyama, May 11 2025
  • Sage
    # uses[MLPower from A326327]
for n in (0..6): print(MLPower(2, n, 9))

Formula

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

A381512 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = (2*n+k)!/k! * [x^(2*n+k)] sinh(x)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 10, 16, 1, 0, 1, 20, 91, 64, 1, 0, 1, 35, 336, 820, 256, 1, 0, 1, 56, 966, 5440, 7381, 1024, 1, 0, 1, 84, 2352, 24970, 87296, 66430, 4096, 1, 0, 1, 120, 5082, 90112, 631631, 1397760, 597871, 16384, 1, 0, 1, 165, 10032, 273988, 3331328, 15857205, 22368256, 5380840, 65536, 1, 0
Offset: 0

Views

Author

Seiichi Manyama, May 11 2025

Keywords

Examples

			Square array begins:
  1, 1,    1,     1,       1,        1, ...
  0, 1,    4,    10,      20,       35, ...
  0, 1,   16,    91,     336,      966, ...
  0, 1,   64,   820,    5440,    24970, ...
  0, 1,  256,  7381,   87296,   631631, ...
  0, 1, 1024, 66430, 1397760, 15857205, ...

Crossrefs

Columns k=0..7 give A000007, A000012, A000302, A002452(n+1), A166984, A002453, 4^n * A002451(n), A381513.
Main diagonal gives A383837.
Cf. A136630, A160562, A286899.

Programs

  • PARI
    a(n, k) = (2*n+k)!/k!*polcoef(sinh(x+x*O(x^(2*n+k)))^k, 2*n+k);

Formula

G.f. of column k: 1/Product_{j=0..floor(k/2)} (1 - (k-2*j)^2*x).
A(n,k) = k^2 * A(n-1,k) + A(n,k-2) for k > 1.
A(n,k) = (1/(2^k*k!)) * Sum_{j=0..k} (-1)^j * (k-2*j)^(2*n+k) * binomial(k,j).

A290772 Number of cyclic Gray codes of length 2n which include all-0 bit sequence and use the least possible number of bits.

Original entry on oeis.org

1, 2, 24, 12, 2640, 7536, 9408, 2688, 208445760, 1082368560, 4312566720, 12473296800, 24050669760, 27034640640, 13900259520, 1813091520
Offset: 1

Views

Author

Ashis Kumar Mal, Aug 10 2017

Keywords

Comments

From Andrey Zabolotskiy, Aug 23 2017: (Start)
The smallest number of bits needed is ceiling(log_2(n)). For larger number of bits, more Gray codes exist. Cyclic Gray codes of odd lengths do not exist, hence only even lengths are considered.
A003042 is a subsequence: A003042(n+1) = a(2^n).
a(n) is also the number of self-avoiding directed cycles of length 2n on a cube of the least possible dimension starting from the origin.
(End)

Examples

			Let n=3, so we count codes of length 6. Then at least 3 bits are needed to have such a code. There are a(3)=24 3-bit cyclic Gray codes of length 6:
000, 001, 011, 010, 110, 100
000, 001, 011, 111, 110, 100
000, 001, 011, 111, 110, 010
000, 001, 011, 111, 101, 100
000, 001, 101, 100, 110, 010
000, 001, 101, 111, 110, 100
000, 001, 101, 111, 110, 010
000, 001, 101, 111, 011, 010
000, 010, 011, 001, 101, 100
000, 010, 011, 111, 110, 100
000, 010, 011, 111, 101, 100
000, 010, 011, 111, 101, 001
000, 010, 110, 111, 101, 100
000, 010, 110, 111, 101, 001
000, 010, 110, 111, 011, 001
000, 010, 110, 100, 101, 001
000, 100, 101, 111, 110, 010
000, 100, 101, 111, 011, 010
000, 100, 101, 111, 011, 001
000, 100, 101, 001, 011, 010
000, 100, 110, 111, 101, 001
000, 100, 110, 111, 011, 010
000, 100, 110, 111, 011, 001
000, 100, 110, 010, 011, 001

Links

Crossrefs

Cf. A003042, A286899, A350784.

Programs

  • Python
    from math import log2, ceil
def cyclic_gray(nb, n, a):
    if len(a) == n:
        if bin(a[-1]).count('1') == 1:
            return 1
        return 0
    r = 0
    for i in range(nb):
        x = a[-1] ^ (1<Andrey Zabolotskiy, Aug 23 2017

Extensions

a(7)-a(8) and name from Andrey Zabolotskiy, Aug 23 2017
a(9)-a(13) from Ashis Kumar Mal, Sep 02 2017
a(14)-a(16) from Thomas König, Jan 22 2022
Showing 1-3 of 3 results.

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