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 1-10 of 19 results. Next

A364446 Odd bisection of A097514.

Original entry on oeis.org

1, 2, 17, 205, 3876, 99585, 3313117, 138046940, 6974868139, 419104459913, 29405917751526, 2376498296500063, 218615700758838253, 22667167720595002186, 2626657814273218158997, 337692419653329329932633, 47859496337287704749354668
Offset: 0

Views

Author

Karol A. Penson, Jul 25 2023

Keywords

Comments

Closed-form expression in terms of known functions.
a(n) is the number of partitions of a (2n+1)-set without blocks of size 2. - Alois P. Heinz, Jul 25 2023

Crossrefs

Cf. A097514.

Programs

  • Maple
    # Maple program 1:
Digits:=48;
a:= proc(n) round(evalf(sum(p^(2*n + 1)*hypergeom([-n, -n - 1/2],
       [ ], -2/p^2)/p!, p = 1 .. infinity)/exp(1)));
    end:
seq(a(n),n=0..16);
# Alternative formula in terms of generalized Laguerre
# polynomials LaguerreL(n,b,z):
# Maple program 2:
Digits:=48;
a:= proc(n) round(evalf(sum(factor(expand(p^(2*n+1)*n!*
      (-2/p^2)^n*LaguerreL(n,1/2,p^2/2)))/p!,p=1..infinity)/exp(1)));
    end:
seq(a(n),n=0..16);
# third Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(`if`(
       j=2, 0, b(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
a:= n-> b(2*n+1):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 25 2023
  • Mathematica
    b[n_] := b[n] = If[n == 0, 1,
   Sum[If[j == 2, 0, b[n-j]*Binomial[n-1, j-1]], {j, 1, n}]];
a[n_] := b[2n+1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 03 2024, after Alois P. Heinz *)
  • PARI
    my(N=44,x='x+O('x^N)); v=Vec(serlaplace(exp(exp(x)-1-x^2/2))); vector(#v\2,n,v[2*n]) \\ Joerg Arndt, Jul 26 2023

Formula

a(n) = Sum_{p >= 1} (p^(2*n + 1)*hypergeom([-n, -n - 1/2], [ ], -2/p^2)/p!) / exp(1).
a(n) = (2*n+1)! * [x^(2*n+1)] exp(exp(x)-1-x^2/2). - Alois P. Heinz, Jul 25 2023

A111723 Number of partitions of an n-set with an odd number of blocks of size 1.

Original entry on oeis.org

1, 0, 4, 4, 31, 86, 449, 1968, 10420, 56582, 333235, 2069772, 13606113, 94065232, 682242552, 5175100432, 40954340995, 337362555010, 2886922399649, 25616738519384, 235313456176512, 2234350827008170, 21899832049913999, 221292603495494488, 2302631998398438321
Offset: 1

Views

Author

Vladeta Jovovic, Nov 17 2005

Keywords

Links

Crossrefs

Cf. A097514, A113235, A063083, A062282, A111724, A111752, A111753.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, t, add(b(n-j,
      `if`(j=1, 1-t, t))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..30);  # Alois P. Heinz, May 10 2016
  • Mathematica
    Rest[ Range[0, 23]! CoefficientList[ Series[ Sinh[x]Exp[Exp[x] - 1 - x], {x, 0, 23}], x]] (* Robert G. Wilson v *)
  • Python
    from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, t):
    return t if n==0 else sum(b(n - j, (1 - t if j==1 else t))*binomial(n - 1, j - 1) for j in range(1, n + 1))
def a(n):
    return b(n, 0)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 10 2017

Formula

E.g.f.: sinh(x)*exp(exp(x)-1-x).
More generally, e.g.f. for number of partitions of an n-set with an odd number of blocks of size k is sinh(x^k/k!)*exp(exp(x)-1-x^k/k!).

Extensions

More terms from Robert G. Wilson v, Nov 22 2005

A111724 Number of partitions of an n-set with an even number of blocks of size 1.

Original entry on oeis.org

0, 2, 1, 11, 21, 117, 428, 2172, 10727, 59393, 345335, 2143825, 14038324, 96834090, 700715993, 5305041715, 41910528809, 344714251149, 2945819805408, 26107419715988, 239556359980239, 2272364911439153, 22252173805170347, 224666265799310801, 2335958333831561032
Offset: 1

Views

Author

Vladeta Jovovic, Nov 17 2005

Keywords

Links

Crossrefs

Cf. A097514, A113235, A063083, A062282, A111723, A111752, A111753.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, t, add(b(n-j,
      `if`(j=1, 1-t, t))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=1..30);  # Alois P. Heinz, May 10 2016
  • Mathematica
    Rest[ Range[0, 24]! CoefficientList[ Series[ Cosh[x]Exp[Exp[x] - 1 - x], {x, 0, 23}], x]] (* Robert G. Wilson v *)
  • Python
    from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, t): return t if n==0 else sum(b(n - j, (1 - t if j==1 else t))*binomial(n - 1, j - 1) for j in range(1, n + 1))
def a(n): return b(n, 1)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 10 2017

Formula

E.g.f.: cosh(x)*exp(exp(x)-1-x).
More generally, e.g.f. for number of partitions of an n-set with an even number of blocks of size k is cosh(x^k/k!)*exp(exp(x)-1-x^k/k!).

Extensions

More terms from Robert G. Wilson v, Nov 22 2005

A113235 Number of partitions of {1,..,n} into any number of lists of size not equal to 2, where a list means an ordered subset, cf. A000262.

Original entry on oeis.org

1, 1, 1, 7, 49, 301, 2281, 21211, 220417, 2528569, 32014801, 442974511, 6638604721, 107089487077, 1849731389689, 34051409587651, 665366551059841, 13751213558077681, 299644435399909537, 6864906328749052759, 164941239260973870001, 4146673091958686331421
Offset: 0

Views

Author

Karol A. Penson, Oct 19 2005

Keywords

Links

Crossrefs

Cf. A000262, A000266, A005251, A052845, A097514.
This sequence, A113236 and A113237 all describe the same type of mathematical structure: lists with some restrictions.

Programs

  • Magma
    I:=[1, 1, 7, 49]; [1] cat [n le 4 select I[n] else (2*n-1)*Self(n -1) - (n-1)*n*Self(n-2) +4*(n-1)*(n-2)*Self(n-3) -2*(n-1)*(n-2)*(n-3)* Self(n-4): n in [1..30]]; // G. C. Greubel, May 16 2018
  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=[1, $3..n]))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016
  • Mathematica
    f[n_] := n!*Sum[(-1)^k*LaguerreL[n - 2*k, -1, -1]/k!, {k, 0, Floor[n/2]}]; Table[ f[n], {n, 0, 19}]
Range[0, 19]!*CoefficientList[ Series[ Exp[x*(1 - x + x^2)/(1 - x)], {x, 0, 19}], x] (* Robert G. Wilson v, Oct 21 2005 *)
  • PARI
    m=30; v=concat([1,1,7,49], vector(m-4)); for(n=5, m, v[n]=(2*n-1)*v[n-1]-(n-1)*n*v[n-2]+4*(n-1)*(n-2)*v[n-3]-2*(n-1)*(n-2)*(n-3)*v[n -4]); concat([1], v) \\ G. C. Greubel, May 16 2018
  • PARI
    x='x+O('x^99); Vec(serlaplace(exp(x*(1-x+x^2)/(1-x)))) \\ Altug Alkan, May 17 2018

Formula

Expression as a sum involving generalized Laguerre polynomials, in Mathematica notation: a(n)=n!*Sum[(-1)^k*LaguerreL[n - 2*k, -1, -1]/k!, {k, 0, Floor[n/2]}], n=0, 1... .
E.g.f.: exp(x*(1-x+x^2)/(1-x)).
From Vaclav Kotesovec, Nov 13 2017: (Start)
a(n) = (2*n - 1)*a(n-1) - (n-1)*n*a(n-2) + 4*(n-2)*(n-1)*a(n-3) - 2*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ exp(-3/2 + 2*sqrt(n) - n) * n^(n-1/4) / sqrt(2) * (1 + 91/(48*sqrt(n))).
(End)

A337058 E.g.f.: 1 / (2 + x^2/2 - exp(x)).

Original entry on oeis.org

1, 1, 2, 7, 33, 191, 1323, 10711, 99151, 1032385, 11943003, 151979213, 2109829857, 31730171539, 513903517585, 8917723105003, 165065061436755, 3246274767649637, 67598797715175999, 1485845872704318265, 34378343609138619685, 835190283258080561671
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 13 2020

Keywords

Crossrefs

Cf. A000670, A032032, A097514, A102233, A337059.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[1/(2 + x^2/2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 21}]

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=3..n} binomial(n,k) * a(n-k).

A343664 Number of partitions of an n-set without blocks of size 4.

Original entry on oeis.org

1, 1, 2, 5, 14, 47, 173, 702, 3125, 14910, 76495, 418035, 2418397, 14791597, 95093612, 641094695, 4521228732, 33250447919, 254585084539, 2024995604762, 16702070759557, 142642458681486, 1259387604241013, 11479967000116911, 107910143688962037, 1044735841257587203, 10407104137208385924
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Links

Crossrefs

Cf. A000110, A000296, A027338, A097514, A124504, A343665, A343666, A343667, A343668, A343669.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=4, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..26);  # Alois P. Heinz, Apr 25 2021
  • Mathematica
    nmax = 26; CoefficientList[Series[Exp[Exp[x] - 1 - x^4/4!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 4 k]/((n - 4 k)! k! (4!)^k), {k, 0, Floor[n/4]}], {n, 0, 26}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 4, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 26}]

Formula

E.g.f.: exp(exp(x) - 1 - x^4/4!).
a(n) = n! * Sum_{k=0..floor(n/4)} (-1)^k * Bell(n-4*k) / ((n-4*k)! * k! * (4!)^k).

A343665 Number of partitions of an n-set without blocks of size 5.

Original entry on oeis.org

1, 1, 2, 5, 15, 51, 197, 835, 3860, 19257, 102997, 586170, 3535645, 22496437, 150454918, 1054235150, 7718958995, 58905868192, 467530598983, 3851775136517, 32881385742460, 290387471713872, 2649226725182823, 24934118754400767, 241809265181914545, 2413608066257526577
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Crossrefs

Cf. A000110, A000296, A027339, A097514, A124504, A343664, A343666, A343667, A343668, A343669.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=5, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021
  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^5/5!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 5 k]/((n - 5 k)! k! (5!)^k), {k, 0, Floor[n/5]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 5, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

Formula

E.g.f.: exp(exp(x) - 1 - x^5/5!).
a(n) = n! * Sum_{k=0..floor(n/5)} (-1)^k * Bell(n-5*k) / ((n-5*k)! * k! * (5!)^k).

A343666 Number of partitions of an n-set without blocks of size 6.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 202, 870, 4084, 20727, 112825, 654546, 4026487, 26145511, 178550986, 1278168860, 9564026947, 74615547996, 605593775899, 5103054929621, 44564754448972, 402677613100491, 3759094788129312, 36205919126040190, 359340174509911325, 3670825700549853053
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Crossrefs

Cf. A000110, A000296, A027340, A097514, A124504, A343664, A343665, A343667, A343668, A343669.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=6, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021
  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^6/6!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 6 k]/((n - 6 k)! k! (6!)^k), {k, 0, Floor[n/6]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 6, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

Formula

E.g.f.: exp(exp(x) - 1 - x^6/6!).
a(n) = n! * Sum_{k=0..floor(n/6)} (-1)^k * Bell(n-6*k) / ((n-6*k)! * k! * (6!)^k).

A343667 Number of partitions of an n-set without blocks of size 7.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 876, 4132, 21075, 115375, 673620, 4172413, 27296089, 187891174, 1356343385, 10238632307, 80615222404, 660560758879, 5621465069117, 49594663447612, 452846969975391, 4273130715906123, 41612346388251187, 417668648929556073, 4315893703814296053
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Crossrefs

Cf. A000110, A000296, A027341, A097514, A124504, A343664, A343665, A343666, A343668, A343669.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=7, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021
  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^7/7!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 7 k]/((n - 7 k)! k! (7!)^k), {k, 0, Floor[n/7]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 7, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

Formula

E.g.f.: exp(exp(x) - 1 - x^7/7!).
a(n) = n! * Sum_{k=0..floor(n/7)} (-1)^k * Bell(n-7*k) / ((n-7*k)! * k! * (7!)^k).

A343668 Number of partitions of an n-set without blocks of size 8.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4139, 21138, 115885, 677745, 4206172, 27577513, 190289713, 1377315050, 10426866782, 82350895629, 677003941219, 5781485704892, 51193839084907, 469251258854001, 4445769329586348, 43475305461354931, 438270620701587657, 4549243731200717053
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Crossrefs

Cf. A000110, A000296, A027342, A097514, A124504, A343664, A343665, A343666, A343667, A343669.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=8, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021
  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^8/8!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 8 k]/((n - 8 k)! k! (8!)^k), {k, 0, Floor[n/8]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 8, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

Formula

E.g.f.: exp(exp(x) - 1 - x^8/8!).
a(n) = n! * Sum_{k=0..floor(n/8)} (-1)^k * Bell(n-8*k) / ((n-8*k)! * k! * (8!)^k).
Showing 1-10 of 19 results. Next

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