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.

Previous Showing 11-19 of 19 results.

A343669 Number of partitions of an n-set without blocks of size 9.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21146, 115965, 678460, 4212497, 27633712, 190795218, 1381942530, 10470109267, 82764226404, 681048663329, 5822029128397, 51610194855972, 473631475252041, 4492967510009533, 43996047374513046, 444151309687221889, 4617189912288741028
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Crossrefs

Cf. A000110, A000296, A027343, A097514, A124504, A343664, A343665, A343666, A343667, A343668.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=9, 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^9/9!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 9 k]/((n - 9 k)! k! (9!)^k), {k, 0, Floor[n/9]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 9, 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^9/9!).
a(n) = n! * Sum_{k=0..floor(n/9)} (-1)^k * Bell(n-9*k) / ((n-9*k)! * k! * (9!)^k).

A327885 Number of set partitions of [n] such that at least one of the block sizes is 2.

Original entry on oeis.org

0, 0, 1, 3, 9, 35, 150, 672, 3269, 17271, 97155, 578985, 3654750, 24331320, 170074177, 1244911605, 9520843575, 75890001665, 629104453236, 5413637745144, 48277814341765, 445463898405225, 4246785220234557, 41775507558584283, 423516880995944532
Offset: 0

Views

Author

Alois P. Heinz, Sep 28 2019

Keywords

Examples

			a(2) = 1: 12.
a(3) = 3: 12|3, 13|2, 1|23.
a(4) = 9: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
a(5) = 35: 123|45, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 12|3|45, 12|3|4|5, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 13|2|45, 13|2|4|5, 145|23, 14|235, 14|23|5, 15|234, 15|23|4, 1|23|45, 1|23|4|5, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|24|35, 1|24|3|5, 15|2|34, 1|25|34, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.

Links

Crossrefs

Column k=2 of A327884.
Cf. A000110, A097514.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1, add(
      `if`(j=k, 0, b(n-j, k)*binomial(n-1, j-1)), j=1..n))
    end:
a:= n-> b(n, 0)-b(n, 2):
seq(a(n), n=0..27);
  • Mathematica
    b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[If[j == k, 0, b[n - j, k]* Binomial[n - 1, j - 1]], {j, n}]];
a[n_] := b[n, 0] - b[n, 2];
a /@ Range[0, 27] (* Jean-François Alcover, May 04 2020, after Maple *)

Formula

E.g.f.: exp(exp(x)-1) - exp(exp(x)-1-x^2/2).
a(n) = A000110(n) - A097514(n).

A364450 Number of partitions of [n] without prime sized blocks.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 17, 43, 135, 536, 2262, 9109, 40119, 198069, 1057149, 5656915, 31937212, 191032078, 1218669125, 7948119483, 54117500635, 381631039690, 2828205076600, 21507011811289, 169880627954541, 1377653319819302, 11620433411120653, 100417638302823210
Offset: 0

Views

Author

Alois P. Heinz, Jul 25 2023

Keywords

Examples

			a(4) = 2: 1|2|3|4, 1234.

Links

Crossrefs

Cf. A000040, A097514, A190476, A329944.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
      isprime(j), 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..30);

Formula

E.g.f.: exp(exp(x)-1-Sum_{p in primes} x^p/p!).

A113237 E.g.f.: exp(x*(1 - x^3 + x^4)/(1-x)).

Original entry on oeis.org

1, 1, 3, 13, 49, 381, 2971, 26713, 291873, 3262969, 41245651, 569262981, 8433896593, 136060620853, 2342471665899, 42987065380561, 838321137046081, 17272648375895793, 375413770580941603, 8579701021461918589, 205637099039964274161, 5158188565847339152621
Offset: 0

Views

Author

Karol A. Penson, Oct 19 2005

Keywords

Comments

Number of partitions of {1,..,n} into any number of lists of size not equal to 4, where a list means an ordered subset, cf. A000262.

Links

Crossrefs

Cf. A000262, A052845, A097514, A113235, A113236.

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!), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016
  • Mathematica
    f[n_] := n!*Sum[(-1)^k*LaguerreL[n - 4*k, -1, -1]/k!, {k, 0, Floor[n/4]}]; Table[ f[n], {n, 0, 19}]
Range[0, 19]!* CoefficientList[ Series[ Exp[x*(1 - x^3 + x^4)/(1 - x)], {x, 0, 19}], x] (* Robert G. Wilson v, Oct 21 2005 *)

Formula

Expression as a sum involving generalized Laguerre polynomials, in Mathematica notation: a(n)=n!*Sum[(-1)^k*LaguerreL[n - 4*k, -1, -1]/k!, {k, 0, Floor[n/4]}], n=0, 1....
Recurrence: a(n) = (2*n-1)*a(n-1) - (n-2)*(n-1)*a(n-2) - 4*(n-3)*(n-2)*(n-1)*a(n-4) + 8*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5) - 4*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6). - Vaclav Kotesovec, Jun 24 2013
a(n) ~ n^(n-1/4)*exp(-3/2+2*sqrt(n)-n)/sqrt(2) * (1 + 187/(48*sqrt(n))). - Vaclav Kotesovec, Jun 24 2013

A124498 Triangle read by rows: T(n,k) is the number of set partitions of the set {1,2,...,n} containing k blocks of size 2 (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 6, 3, 17, 20, 15, 53, 90, 45, 15, 205, 357, 210, 105, 871, 1484, 1260, 420, 105, 3876, 7380, 6426, 2520, 945, 18820, 39195, 33390, 18900, 4725, 945, 99585, 213180, 202950, 117810, 34650, 10395, 558847, 1242120, 1293435, 734580, 311850
Offset: 0

Views

Author

Emeric Deutsch, Nov 05 2006

Keywords

Comments

Row n contains 1+floor(n/2) terms. Row sums yield the Bell numbers A000110. T(n,0)=A097514(n). Sum(k*T(n,k), k=0..floor(n/2))=A105479(n+1).

Examples

			T(4,1)=6 because we have 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3 and 1|2|34.
Triangle T(n,k) begins:
:      1;
:      1;
:      1,       1;
:      2,       3;
:      6,       6,       3;
:     17,      20,      15;
:     53,      90,      45,     15;
:    205,     357,     210,    105;
:    871,    1484,    1260,    420,    105;
:   3876,    7380,    6426,   2520,    945;
:  18820,   39195,   33390,  18900,   4725,   945;
:  99585,  213180,  202950, 117810,  34650, 10395;
: 558847, 1242120, 1293435, 734580, 311850, 62370, 10395;

Links

Crossrefs

Cf. A000110, A097514, A105479, A124503.
T(2n,n) gives A001147.

Programs

  • Maple
    G:=exp(exp(z)-1+(t-1)*z^2/2): Gser:=simplify(series(G,z=0,16)): for n from 0 to 13 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 13 do seq(coeff(P[n],t,k),k=0..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(i=2, x^j, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Mar 08 2015
  • Mathematica
    d=Exp[Exp[x]-x^2/2!-1]; f[list_] := Select[list,#>0&]; Map[f, Transpose[Table[Range[0,12]! CoefficientList[Series[ x^(2k)/(k! 2!^k) *d, {x,0,12}], x], {k,0,5}]]]//Flatten (* Geoffrey Critzer, Nov 30 2011 *)

Formula

E.g.f.: exp(exp(z)-1+(t-1)z^2/2).
Generally the e.g.f. for set partitions containing k blocks of size p is: G(z,t) = exp(exp(z)-1+(t-1)z^p/p!) - Geoffrey Critzer, Nov 30 2011

A337062 E.g.f.: exp(1 + x^2/2 - exp(x)).

Original entry on oeis.org

1, -1, 1, -2, 4, -7, 21, -51, 113, -498, 1088, -3335, 21407, -14653, 232389, -1275288, -3636526, -44468245, -7468609, 700603965, 12178055777, 67189448344, 175549544778, -2432123216941, -36279392911507, -287078642854853, -945866835928323
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 13 2020

Keywords

Crossrefs

Cf. A000587, A001147, A097514, A293037, A293038.

Programs

  • Mathematica
    nmax = 26; CoefficientList[Series[Exp[1 + x^2/2 - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] - Sum[Binomial[n - 1, k - 1] a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 26}]
Table[Sum[Binomial[n, 2 k] (2 k - 1)!! BellB[n - 2 k, -1], {k, 0, Floor[n/2]}], {n, 0, 26}]

Formula

a(0) = 1; a(n) = -a(n-1) - Sum_{k=3..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k) * (2*k-1)!! * A000587(n-2*k).

A343671 Number of partitions of an n-set without blocks of size 10.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115974, 678559, 4213465, 27643007, 190884307, 1382802389, 10478516523, 82847813908, 681895648039, 5830788687491, 51702731250650, 474630475600569, 4503991075480297, 44120379612630694, 445584481578266277, 4634070027874688433
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Crossrefs

Cf. A000110, A000296, A027344, A097514, A124504, A343664, A343665, A343666, A343667, A343668, A343669.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
       j=10, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 25 2023
  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^10/10!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 10 k]/((n - 10 k)! k! (10!)^k), {k, 0, Floor[n/10]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 10, 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^10/10!).
a(n) = n! * Sum_{k=0..floor(n/10)} (-1)^k * Bell(n-10*k) / ((n-10*k)! * k! * (10!)^k).

A346271 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( BesselI(0,2*sqrt(x)) - 1 - x^2 / 4 ).

Original entry on oeis.org

1, 1, 2, 7, 41, 346, 3807, 53747, 952275, 20362552, 515112983, 15277888693, 523304644304, 20415373547609, 900219731675981, 44533809102813206, 2451041479421900803, 149140880201760643360, 9982798939295116151967, 731215136812226462200109, 58333374310397488522052976
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 12 2021

Keywords

Crossrefs

Cf. A023998, A061696, A097514, A346272.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]] - 1 - x^2/4], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = n a[n - 1] + (1/n) Sum[Binomial[n, k]^2 k a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 20}]

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( x + Sum_{n>=3} x^n / (n!)^2 ).
a(0) = 1; a(n) = n * a(n-1) + (1/n) * Sum_{k=3..n} binomial(n,k)^2 * k * a(n-k).

A360991 Expansion of e.g.f. exp(exp(x) - 1 + x^2/2).

Original entry on oeis.org

1, 1, 3, 8, 30, 117, 533, 2599, 13919, 79620, 487810, 3167265, 21744187, 157020697, 1189321019, 9417789650, 77774264012, 668233623419, 5961395449795, 55117233908411, 527263186773227, 5210880621612366, 53130216638022540, 558176360466846439
Offset: 0

Views

Author

Seiichi Manyama, Mar 14 2023

Keywords

Crossrefs

Cf. A097514, A355337, A361489.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(exp(x)-1+x^2/2)))

Formula

a(0) = a(1) = 1; a(n) = (n-1) * a(n-2) + Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
Previous Showing 11-19 of 19 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.