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 1-5 of 5 results.

A275422 Number A(n,k) of set partitions of [n] such that k is a multiple of each block size; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 1, 15, 1, 1, 1, 4, 1, 52, 1, 1, 2, 2, 10, 1, 203, 1, 1, 1, 4, 5, 26, 1, 877, 1, 1, 2, 1, 11, 11, 76, 1, 4140, 1, 1, 1, 5, 1, 31, 31, 232, 1, 21147, 1, 1, 2, 1, 14, 2, 106, 106, 764, 1, 115975, 1, 1, 1, 4, 1, 46, 7, 372, 337, 2620, 1, 678570
Offset: 0

Views

Author

Alois P. Heinz, Jul 27 2016

Keywords

Examples

			A(5,3) = 11: 123|4|5, 124|3|5, 125|3|4, 134|2|5, 135|2|4, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345, 1|2|3|4|5.
A(4,4) = 11: 1234, 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
A(6,5) = 7: 12345|6, 12346|5, 12356|4, 12456|3, 13456|2, 1|23456, 1|2|3|4|5|6.
Square array A(n,k) begins:
:    1, 1,   1,   1,    1,  1,    1, 1,    1, ...
:    1, 1,   1,   1,    1,  1,    1, 1,    1, ...
:    2, 1,   2,   1,    2,  1,    2, 1,    2, ...
:    5, 1,   4,   2,    4,  1,    5, 1,    4, ...
:   15, 1,  10,   5,   11,  1,   14, 1,   11, ...
:   52, 1,  26,  11,   31,  2,   46, 1,   31, ...
:  203, 1,  76,  31,  106,  7,  167, 1,  106, ...
:  877, 1, 232, 106,  372, 22,  659, 2,  372, ...
: 4140, 1, 764, 337, 1499, 57, 2836, 9, 1500, ...

Links

Crossrefs

Columns k=0-10 give: A000110, A000012, A000085, A190865, A190452, A275423, A275424, A275425, A275426, A275427, A275428.
Main diagonal gives A275429.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, add(
      `if`(j>n, 0, A(n-j, k)*binomial(n-1, j-1)), j=
      `if`(k=0, 1..n, numtheory[divisors](k))))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);
  • Mathematica
    A[n_, k_] := A[n, k] = If[n==0, 1, Sum[If[j>n, 0, A[n-j, k]*Binomial[n-1, j - 1]], {j, If[k==0, Range[n], Divisors[k]]}]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)

Formula

E.g.f. for column k>0: exp(Sum_{d|k} x^d/d!), for k=0: exp(exp(x)-1).

A351931 Expansion of e.g.f. exp(x - x^5/120).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, -5, -20, -55, -125, -125, 925, 7525, 34750, 124125, 249250, -1013375, -14708875, -97413875, -477236375, -1443329375, 3466472500, 91499089375, 804081585000, 5030009685625, 20366827624375, -23484049500625, -1391395435656875, -15503027252406875
Offset: 0

Views

Author

Seiichi Manyama, Feb 26 2022

Keywords

Crossrefs

Cf. A351929, A351930.
Cf. A275423, A351906.

Programs

  • Mathematica
    m = 28; Range[0, m]! * CoefficientList[Series[Exp[x - x^5/5!], {x, 0, m}], x] (* Amiram Eldar, Feb 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x-x^5/5!)))
  • PARI
    a(n) = n!*sum(k=0, n\5, (-1/5!)^k*binomial(n-4*k, k)/(n-4*k)!);
  • PARI
    a(n) = if(n<5, 1, a(n-1)-binomial(n-1, 4)*a(n-5));

Formula

a(n) = n! * Sum_{k=0..floor(n/5)} (-1/5!)^k * binomial(n-4*k,k)/(n-4*k)!.
a(n) = a(n-1) - binomial(n-1,4) * a(n-5) for n > 4.

A351935 Expansion of e.g.f. exp(x / (1 - x^4/24)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 31, 106, 281, 1261, 10711, 71611, 350461, 1808236, 17037021, 170285116, 1293714241, 8653175441, 84433291741, 1063629264781, 11218379358721, 97926941650546, 1021280770603171, 14623420493573046, 197153396050112041, 2190425085571083901
Offset: 0

Views

Author

Seiichi Manyama, Feb 26 2022

Keywords

Crossrefs

Cf. A000262, A351933, A351934, A351936.
Cf. A275423.

Programs

  • Mathematica
    m = 25; Range[0, m]! * CoefficientList[Series[Exp[x/(1 - x^4/24)], {x, 0, m}], x] (* Amiram Eldar, Feb 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x^4/24))))
  • PARI
    a(n) = if(n<5, 1, sum(k=0, (n-1)\4, (4*k+1)!/24^k*binomial(n-1, 4*k)*a(n-1-4*k)));

Formula

a(n) = Sum_{k=0..floor((n-1)/4)} (4*k+1)!/24^k * binomial(n-1,4*k) * a(n-1-4*k) for n > 4.
a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-3*k-1,k)/(24^k * (n-4*k)!). - Seiichi Manyama, Jun 08 2024

A351932 Number of set partitions of [n] such that block sizes are either 1 or 4.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 36, 106, 442, 1786, 6106, 23596, 120836, 631632, 2854216, 13590396, 81258556, 510768316, 2839808572, 16008902296, 108643656136, 787965516416, 5161270717296, 33513036683512, 253407796702776, 2065728484459576, 15485032349429176, 113510664648701776
Offset: 0

Views

Author

Seiichi Manyama, Feb 26 2022

Keywords

Crossrefs

Cf. A000085, A190865, A275423, A275425.
Cf. A190452, A190875, A351930.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1,
     `if`(n<4, 0, a(n-4)*binomial(n-1, 3))+a(n-1))
    end:
seq(a(n), n=0..28);  # Alois P. Heinz, Feb 26 2022
seq(round(evalf(hypergeom([-n/4,(1-n)/4,(2-n)/4,(3-n)/4],[],32/3))),n=0..28);  # Karol A. Penson, Jul 28 2023
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x+x^4/4!)))
  • PARI
    a(n) = n!*sum(k=0, n\4, 1/4!^k*binomial(n-3*k, k)/(n-3*k)!);
  • PARI
    a(n) = if(n<4, 1, a(n-1)+binomial(n-1, 3)*a(n-4));

Formula

E.g.f.: exp(x + x^4/24).
a(n) = n! * Sum_{k=0..floor(n/4)} (1/24)^k * binomial(n-3*k,k)/(n-3*k)!.
a(n) = a(n-1) + binomial(n-1,3) * a(n-4) for n > 3.
a(n) = hypergeom([-n/4,(1-n)/4,(2-n)/4,(3-n)/4],[],32/3), Karol A. Penson, Jul 28 2023.

A362336 a(n) = n! * Sum_{k=0..floor(n/5)} (n/120)^k /(k! * (n-5*k)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 37, 148, 449, 1135, 15121, 172789, 1207009, 6106816, 24748725, 510855346, 8524169473, 84981641837, 602009065729, 3357322881625, 93871272204481, 2059974308136466, 26683062726210661, 243032907824598816, 1725747644222610625
Offset: 0

Views

Author

Seiichi Manyama, Apr 16 2023

Keywords

Links

Crossrefs

Cf. A362173, A362317.
Cf. A275423, A362319, A362323.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((-24*lambertw(-x^5/24))^(1/5))/(1+lambertw(-x^5/24))))

Formula

a(n) = n! * [x^n] exp(x + n*x^5/120).
E.g.f.: exp( ( -24*LambertW(-x^5/24) )^(1/5) ) / (1 + LambertW(-x^5/24)).
Showing 1-5 of 5 results.

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