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 1-4 of 4 results.

A326327 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, 5, -2, 1, 0, -61, 16, -3, 1, 0, 1385, -272, 33, -4, 1, 0, -50521, 7936, -723, 56, -5, 1, 0, 2702765, -353792, 25953, -1504, 85, -6, 1, 0, -199360981, 22368256, -1376643, 64256, -2705, 120, -7, 1, 0, 19391512145, -1903757312, 101031873, -3963904, 134185, -4416, 161, -8, 1
Offset: 0

Views

Author

Peter Luschny, Jul 07 2019

Keywords

Examples

			Array starts:
[0] 1,  0,   0,     0,      0,         0,          0,             0, ... A000007
[1] 1, -1,   5,   -61,   1385,    -50521,    2702765,    -199360981, ... A028296
[2] 1, -2,  16,  -272,   7936,   -353792,   22368256,   -1903757312, ... A000182
[3] 1, -3,  33,  -723,  25953,  -1376643,  101031873,   -9795436563, ... A326328
[4] 1, -4,  56, -1504,  64256,  -3963904,  332205056,  -36246728704, ...
[5] 1, -5,  85, -2705, 134185,  -9451805,  892060285, -108357876905, ...
[6] 1, -6, 120, -4416, 249600, -19781376, 2078100480, -278400270336, ...
       A045944,
Seen as a triangle:
[0] [1]
[1] [0, 1]
[2] [0, -1,      1]
[3] [0, 5,       -2,      1]
[4] [0, -61,     16,      -3,    1]
[5] [0, 1385,    -272,    33,    -4,    1]
[6] [0, -50521,  7936,    -723,  56,    -5, 1]
[7] [0, 2702765, -353792, 25953, -1504, 85, -6, 1]

Links

Crossrefs

Rows: A000007 (row 0), A028296 (row 1), A000182 (row 2), A326328(row 3).
Columns: A045944 (col. 2).
Cf. A326476 (m=2, p>=0), this sequence (m=2, p<=0), A326474 (m=3, p>=0), A326475 (m=3, p<=0).

Programs

  • Mathematica
    cl[m_, p_, len_] := CoefficientList[
   Series[FunctionExpand[MittagLefflerE[m, z]^p], {z, 0, len}], z];
MLPower[m_, 0,  len_] := Table[KroneckerDelta[0, n], {n, 0, len - 1}];
MLPower[m_, n_, len_] := cl[m, n, len - 1] (m Range[0, len - 1])!;
For[n = 0, n < 8, n++, Print[MLPower[2, -n, 8]]]
  • Sage
    def MLPower(m, p, len):
    if p == 0: return [p^k for k in (0..len-1)]
    f = [i/m for i in (1..m-1)]
    h = lambda x: hypergeometric([], f, (x/m)^m)
    g = [v for v in taylor(h(x)^p, x, 0, (len-1)*m).list() if v != 0]
    return [factorial(m*k)*v for (k, v) in enumerate(g)]
for p in (0..6): print(MLPower(2, -p, 9))

A326474 A(n, k) = (m*k)! [x^k] MittagLefflerE(m, x)^n, for m = 3, 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, 22, 3, 1, 0, 1, 170, 63, 4, 1, 0, 1, 1366, 2187, 124, 5, 1, 0, 1, 10922, 59535, 7732, 205, 6, 1, 0, 1, 87382, 1594323, 599548, 18485, 306, 7, 1, 0, 1, 699050, 43033599, 39945364, 2416045, 36126, 427, 8, 1
Offset: 0

Views

Author

Peter Luschny, Jul 08 2019

Keywords

Examples

			Array starts:
[0] 1, 0,   0,     0,       0,          0,            0, ... A000007
[1] 1, 1,   1,     1,       1,          1,            1, ... A000012
[2] 1, 2,  22,   170,    1366,      10922,        87382, ... A007613
[3] 1, 3,  63,  2187,   59535,    1594323,     43033599, ...
[4] 1, 4, 124,  7732,  599548,   39945364,   2556712828, ...
[5] 1, 5, 205, 18485, 2416045,  352060805,  46660373965, ...
[6] 1, 6, 306, 36126, 6673266, 1544907006, 379696000626, ...
      A051874,

Crossrefs

Rows include: A000007, A000012, A007613.
Columns include: A051874.
Cf. A326476 (m=2, p>=0), A326327 (m=2, p<=0), this sequence (m=3, p>=0), A326475 (m=3, p<=0).

Programs

  • Mathematica
    (* The function MLPower is defined in A326327. *)
For[n = 0, n < 8, n++, Print[MLPower[3, n, 8]]]
  • Sage
    # uses[MLPower from A326327]
for n in (0..6): print(MLPower(3, n, 9))

A326475 A(n, k) = (m*k)! [x^k] MittagLefflerE(m, x)^(-n), for m = 3, n >= 0, k >= 0; square array read by descending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 19, -2, 1, 0, -1513, 58, -3, 1, 0, 315523, -6218, 117, -4, 1, 0, -136085041, 1630330, -15795, 196, -5, 1, 0, 105261234643, -847053482, 4997781, -31924, 295, -6, 1, 0, -132705221399353, 766492673914, -3042574083, 11840836, -56285, 414, -7, 1
Offset: 0

Views

Author

Peter Luschny, Jul 08 2019

Keywords

Examples

			Array starts:
[0] 1,  0,   0,      0,        0,            0, ... A000007
[1] 1, -1,  19,  -1513,   315523,   -136085041, ... A002115
[2] 1, -2,  58,  -6218,  1630330,   -847053482, ...
[3] 1, -3, 117, -15795,  4997781,  -3042574083, ...
[4] 1, -4, 196, -31924, 11840836,  -8271354004, ...
[5] 1, -5, 295, -56285, 23952055, -18889306805, ...
[6] 1, -6, 414, -90558, 43493598, -38227720446, ...

Crossrefs

Rows: A000007, A002115.
Cf. A326476 (m=2, p>=0), A326327 (m=2, p<=0), A326474 (m=3, p>=0), this sequence (m=3, p<=0).

Programs

  • Mathematica
    (* The function MLPower is defined in A326327. *)
For[n = 0, n < 8, n++, Print[MLPower[3, -n, 8]]]
  • Sage
    # uses[MLPower from A326327]
for n in (0..6): print(MLPower(3, -n, 9))

A381459 a(n) = (2*n)! * [x^(2*n)] cosh(x)^n.

Original entry on oeis.org

1, 1, 8, 183, 8320, 628805, 71172096, 11266376947, 2376282177536, 644092653605769, 218152097885716480, 90283850458537906511, 44828889635978905387008, 26302150870235970074916493, 18001952557737056033350615040, 14215240470695667525160827723915
Offset: 0

Views

Author

Seiichi Manyama, May 11 2025

Keywords

Crossrefs

Main diagonal of A326476.
Cf. A242446.

Programs

  • Mathematica
    Table[(2*n)! * SeriesCoefficient[Cosh[x]^n, {x, 0, 2*n}], {n, 0, 20}] (* Vaclav Kotesovec, May 11 2025 *)
  • PARI
    a(n) = sum(k=0, n, (n-2*k)^(2*n)*binomial(n, k))/2^n;

Formula

a(n) = (1/2^n) * Sum_{k=0..n} (n-2*k)^(2*n) * binomial(n,k).
a(n) ~ c * ((1-2*r)^2 / (2 * r^r * (1-r)^(1-r)))^n * n^(2*n), where r = 0.015817782507793257357841601600685290637088885324182071456255... is the root of the equation (1-2*r)*(log(1-r) - log(r)) = 4 and c = 2*(1 - 2*r) / sqrt(1 + 4*r - 4*r^2) = 1.879106100687674868112932937483753439332007654254262530564... - Vaclav Kotesovec, May 11 2025, updated May 12 2025
Showing 1-4 of 4 results.

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