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.

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A157401 A partition product of Stirling_2 type [parameter k = 1] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 9, 12, 15, 1, 25, 60, 75, 105, 1, 75, 330, 450, 630, 945, 1, 231, 1680, 3675, 4410, 6615, 10395, 1, 763, 9408, 30975, 41160, 52920, 83160, 135135, 1, 2619, 56952, 233415, 489510, 555660, 748440, 1216215
Offset: 1

Views

Author

Peter Luschny, Mar 09 2009, Mar 14 2009

Keywords

Comments

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 1,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A143171.
Same partition product with length statistic is A001497.
Diagonal a(A000217) = A001147.
Row sum is A001515.

Links

Crossrefs

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157402, A157403, A157404, A157405

Formula

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(2*j - 1).

A157402 A partition product of Stirling_2 type [parameter k = 2] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 2, 1, 6, 10, 1, 24, 40, 80, 1, 80, 300, 400, 880, 1, 330, 2400, 3600, 5280, 12320, 1, 1302, 15750, 47600, 55440, 86240, 209440, 1, 5936, 129360, 588000, 837760, 1034880, 1675520, 4188800, 1, 26784, 1146040, 5856480
Offset: 1

Views

Author

Peter Luschny, Mar 09 2009, Mar 14 2009

Keywords

Comments

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 2,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A143172.
Same partition product with length statistic is A004747.
Diagonal a(A000217) = A008544.
Row sum is A015735.

Links

Crossrefs

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157403, A157404, A157405

Formula

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(3*j - 1).

A157403 A partition product of Stirling_2 type [parameter k = 3] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 3, 1, 9, 21, 1, 45, 84, 231, 1, 165, 840, 1155, 3465, 1, 855, 8610, 13860, 20790, 65835, 1, 3843, 64680, 250635, 291060, 460845, 1514205, 1, 21819, 689136, 3969735, 6015240, 7373520, 12113640, 40883535, 1, 114075
Offset: 1

Views

Author

Peter Luschny, Mar 09 2009

Keywords

Comments

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 3,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A143173.
Same partition product with length statistic is A000369.
Diagonal a(A000217) = A008545
Row sum is A016036.

Links

Crossrefs

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157404, A157405

Formula

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(4*j - 1).

A157404 A partition product of Stirling_2 type [parameter k = 4] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 4, 1, 12, 36, 1, 72, 144, 504, 1, 280, 1800, 2520, 9576, 1, 1740, 22320, 37800, 57456, 229824, 1, 8484, 182700, 864360, 1005480, 1608768, 6664896, 1, 57232, 2380896, 16546320, 26276544, 32175360, 53319168, 226606464
Offset: 1

Views

Author

Peter Luschny, Mar 09 2009

Keywords

Comments

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 4,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144267.
Same partition product with length statistic is A011801.
Diagonal a(A000217) = A008546.
Row sum is A028575.

Links

Crossrefs

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157403, A157405

Formula

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(5*j - 1).

A157405 A partition product of Stirling_2 type [parameter k = 5] with biggest-part statistic (triangle read by rows).

Original entry on oeis.org

1, 1, 5, 1, 15, 55, 1, 105, 220, 935, 1, 425, 3300, 4675, 21505, 1, 3075, 47850, 84150, 129030, 623645, 1, 15855, 415800, 2323475, 2709630, 4365515, 415800, 2323475, 2709630, 4365515, 21827575, 1, 123515, 6394080, 51934575
Offset: 0

Views

Author

Peter Luschny, Mar 09 2009, Mar 14 2009

Keywords

Comments

Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 5,
summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A144268.
Same partition product with length statistic is A013988.
Diagonal a(A000217) = A008543.
Row sum is A028844.

Links

Crossrefs

Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157403, A157404

Formula

T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(6*j - 1).

A084708 Number of set partitions up to rotations and reflections.

Original entry on oeis.org

1, 2, 3, 7, 12, 37, 93, 354, 1350, 6351, 31950, 179307, 1071265, 6845581, 46162583, 327731950, 2437753740, 18948599220, 153498350745, 1293123243928, 11306475314467, 102425554299516, 959826755336242, 9290811905391501
Offset: 1

Views

Author

Wouter Meeussen, Jul 02 2003

Keywords

Comments

Combines the symmetry operations of A080107 and A084423.
Equivalently, number of n-bead bracelets using any number of unlabeled (interchangable) colors. - Andrew Howroyd, Sep 25 2017

Examples

			SetPartitions[6] is the first to decompose differently from A084423: 4 cycles of length 1, 2 of 2, 9 of 3, 16 of 6, 6 of 12.
a(7) = 1 + A056357(7) + A056358(7) + A056359(7) + A056360(7) + A056361(7) + 1 = 1 + 8 + 31 + 33 + 16 + 3 + 1 = 93.

Links

Crossrefs

Cf. A080107, A084423, A080510, A002872, A002874, A141003, A036075, A141004, A036077, A152176.

Programs

  • Mathematica
    <A080107 *); Table[{Length[ # ], First[ # ]}&/@ Split[Sort[Length/@Split[Sort[First[Sort[Flatten[ {#, Map[Sort, (#/. i_Integer:>w+1-i), 2]}& @(NestList[Sort[Sort/@(#/. i_Integer :> Mod[i+1, w, 1])]&, #, w]), 1]]]&/@SetPartitions[w]]]]], {w, 1, 10}]
u[0,j_]:=1;u[k_,j_]:=u[k,j]=Sum[Binomial[k-1,i-1]Plus@@(u[k-i,j]#^(i-1)&/@Divisors[j]),{i,k}]; a[n_]:=1/n*Plus@@(EulerPhi[ # ]u[Quotient[n,# ],# ]&/@Divisors[n]); Table[a[n]/2+If[EvenQ[n],u[n/2,2],Sum[Binomial[n/2-1/2,k] u[k,2], {k,0,n/2-1/2}]]/2,{n,40}] (* Wouter Meeussen, Dec 06 2008 *)

Formula

a(n) = (A080107(n)+A084423(n))/2. - Wouter Meeussen and Vladeta Jovovic, Nov 28 2008

Extensions

a(12) from Vladeta Jovovic, Jul 15 2007
More terms from Vladeta Jovovic's formula given in Mathematica line. - Wouter Meeussen, Dec 06 2008

A276961 Number of set partitions of [2n] with largest set of size n.

Original entry on oeis.org

1, 1, 9, 90, 1015, 12978, 187110, 3008148, 53275365, 1028142830, 21426984722, 478684639524, 11394222257054, 287518726261900, 7658231720886900, 214521099685649640, 6299407928673657135, 193373975592937777770, 6189939300880260745050, 206159811915115686404700
Offset: 0

Views

Author

Alois P. Heinz, Sep 22 2016

Keywords

Comments

The blocks are ordered with increasing least elements.
a(0) = 1 by convention.

Examples

			a(1) = 1: 1|2.
a(2) = 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.

Links

Crossrefs

Cf. A000110, A080510, A276923, A297924, A297926, A327884, A328156.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1, add(
      b(n-i, k)*binomial(n-1, i-1), i=1..min(n, k)))
    end:
a:= n-> `if`(n=0, 1, b(2*n, n)-b(2*n, n-1)):
seq(a(n), n=0..20);
  • Mathematica
    b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - i, k]*Binomial[n - 1, i - 1], {i, 1, Min[n, k]}]];
a[n_] := If[n == 0, 1, b[2*n, n] - b[2*n, n - 1]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 20 2018, translated from Maple *)

Formula

a(n) = A080510(2n,n).
a(n) = A327884(2n,n).
a(n) = ceiling(C(2n,n)*(A000110(n)-1/2)). - Ludovic Schwob, Jan 15 2022

A097148 Total sum of maximum block sizes in all partitions of n-set.

Original entry on oeis.org

1, 3, 10, 35, 136, 577, 2682, 13435, 72310, 414761, 2524666, 16239115, 109976478, 781672543, 5814797281, 45155050875, 365223239372, 3070422740989, 26780417126048, 241927307839731, 2260138776632752, 21805163768404127, 216970086170175575, 2224040977932468379
Offset: 1

Views

Author

Vladeta Jovovic, Jul 27 2004

Keywords

Comments

Let M be the infinite lower triangular matrix given by A080510 and v the column vector [1,2,3,...] then M*v=A097148 (this sequence, as column vector). - Gary W. Adamson, Feb 24 2011

Links

Crossrefs

Cf. A028417, A028418, A046746, A006128, A097145-A097147.
Cf. A080510.
Column k=1 of A319375.

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0, m, add(
      b(n-j, max(j, m))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..24);  # Alois P. Heinz, Mar 02 2020, revised May 08 2024
  • Mathematica
    Drop[ Range[0, 22]! CoefficientList[ Series[ Sum[E^(E^x - 1) - E^Sum[x^j/j!, {j, 1, k}], {k, 0, 22}], {x, 0, 22}], x], 1] (* Robert G. Wilson v, Aug 05 2004 *)

Formula

E.g.f.: Sum_{k>=0} (exp(exp(x)-1)-exp(Sum_{j=1..k} x^j/j!)).

Extensions

More terms from Robert G. Wilson v, Aug 05 2004

A327884 Number T(n,k) of set partitions of [n] such that at least one of the block sizes is k or k=0; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 4, 3, 1, 15, 11, 9, 4, 1, 52, 41, 35, 20, 5, 1, 203, 162, 150, 90, 30, 6, 1, 877, 715, 672, 455, 175, 42, 7, 1, 4140, 3425, 3269, 2352, 1015, 280, 56, 8, 1, 21147, 17722, 17271, 13132, 6237, 1890, 420, 72, 9, 1, 115975, 98253, 97155, 76540, 39480, 12978, 3150, 600, 90, 10, 1
Offset: 0

Views

Author

Alois P. Heinz, Sep 28 2019

Keywords

Examples

			T(4,1) = 11: 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
T(4,2) = 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.
T(4,3) = 4: 123|4, 124|3, 134|2, 1|234.
T(4,4) = 1: 1234.
T(5,1) = 41: 1234|5, 1235|4, 123|4|5, 1245|3, 124|3|5, 12|34|5, 125|3|4, 12|35|4, 12|3|45, 12|3|4|5, 1345|2, 134|2|5, 13|24|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 1|2345, 1|234|5, 15|23|4, 1|235|4, 1|23|45, 1|23|4|5, 145|2|3, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|245|3, 1|24|35, 1|24|3|5, 15|2|34, 1|25|34, 1|2|345, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5.
Triangle T(n,k) begins:
      1;
      1,     1;
      2,     1,     1;
      5,     4,     3,     1;
     15,    11,     9,     4,    1;
     52,    41,    35,    20,    5,    1;
    203,   162,   150,    90,   30,    6,   1;
    877,   715,   672,   455,  175,   42,   7,  1;
   4140,  3425,  3269,  2352, 1015,  280,  56,  8, 1;
  21147, 17722, 17271, 13132, 6237, 1890, 420, 72, 9, 1;
  ...

Links

Crossrefs

Columns k=0-3 give: A000110, A000296(n+1), A327885, A328153.
T(2n,n) gives A276961.
Cf. A080510, A327869.

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:
T:= (n, k)-> b(n, 0)-`if`(k=0, 0, b(n, k)):
seq(seq(T(n, k), k=0..n), n=0..11);
  • 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, 1, n}]];
T[n_, k_] := b[n, 0] - If[k == 0, 0, b[n, k]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

Formula

E.g.f. of column k: exp(exp(x)-1) - [k>0] * exp(exp(x)-1-x^k/k!).
T(n,0) - T(n,1) = A000296(n).

A080508 Triangle whose n-th row contains the least set (ordered lexicographically) of n distinct positive integers whose geometric mean is an integer.

Original entry on oeis.org

1, 1, 4, 1, 2, 4, 1, 2, 3, 216, 1, 2, 3, 4, 324, 1, 2, 3, 4, 5, 6075000, 1, 2, 3, 4, 5, 6, 30375000, 1, 2, 3, 4, 5, 6, 7, 750453558750000, 1, 2, 3, 4, 5, 6, 7, 8, 19699405917187500, 1, 2, 3, 4, 5, 6, 7, 8, 9, 459652804734375000, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9652708899421875000
Offset: 1

Views

Author

Amarnath Murthy, Mar 20 2003

Keywords

Comments

The n-th row has the form {1,2,...,n-1,x}, where x is as small as possible.

Examples

			Triangle begins:
  1;
  1, 4;
  1, 2, 4;
  1, 2, 3, 216;
  1, 2, 3,   4, 324;
  1, 2, 3,   4,   5, 6075000;
  ...

Links

Crossrefs

Cf. A080509, A080510, A080504, A080511.

Programs

  • Maple
    f:= proc(n) local F;
  F:= ifactors((n-1)!)[2];
  mul(t[1]^(n-(t[2] mod n)),t=F)
end proc:
f(2):= 4:
seq(op([seq(j,j=1..i-1),f(i)]),i=1..20); # Robert Israel, Nov 04 2018
  • Mathematica
    MapAt[{First@ #, 4 Last@ #} &, Array[Append[Range[# - 1], Apply[Times, Prime@ Range@ PrimePi[# - 1]]^#/(# - 1)!] &, 11], 2] // Flatten (* Michael De Vlieger, Nov 05 2018 *)

Extensions

More terms using A080509 from Michel Marcus, Nov 04 2018
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