A275043 -id:A275043 - cp's OEIS frontend

A061684 Generalized Bell numbers: column 3 of A275043.

1, 1, 5, 64, 1613, 69026, 4566992, 437665649, 57903766797, 10193400044254, 2319001344297830, 665816738235745559, 236563125351122920088, 102303284135845463907107, 53093636013475924370369829, 32666276100771741793923209939, 23573762287735885858839134983437

Offset: 0

N. J. A. Sloane, Jun 18 2001

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..215
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Probably A061698 from the same paper is an erroneous version of this sequence. - Les Reid, Jan 01 2011
Column k=3 of A275043.
Row sums of A061692.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)*binomial(n-1, i-1)/i!^2, i=1..n))
    end:
a:= n-> b(n)*n!^2:
seq(a(n), n=0..20);  # Alois P. Heinz, May 11 2016

b[n_] := b[n] = If[n==0, 1, Sum[b[n-i]*Binomial[n-1, i-1]/i!^2, {i, 1, n}]];
a[n_] := b[n]*n!^2;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 14 2017, after Alois P. Heinz *)

{a(n)=n!^3*polcoeff(exp(sum(m=1, n, x^m/m!^3)+x*O(x^n)), n)} /* Paul D. Hanna, Mar 15 2012 */

G.f.: Sum_{n>=0} a(n)*x^n/n!^3 = exp( Sum_{n>=1} x^n/n!^3 ). [Paul D. Hanna, Mar 15 2012]

a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k)^3 * (n-k) * a(k). - Ilya Gutkovskiy, Jul 12 2020

More terms from Karol A. Penson, Sep 10 2001

A061685 Generalized Bell numbers: column 4 of A275043.

Original entry on oeis.org

1, 1, 9, 298, 25097, 4383626, 1394519922, 738298190981, 608765840524809, 742996254490626106, 1289282092211451157634, 3078466688415490018129781, 9844321075186192301310239858, 41209705023068976933023104392293, 221473347301087557264532943397984133

Offset: 0

N. J. A. Sloane, Jun 18 2001

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..167
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Column k=4 of A275043.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)*binomial(n-1, i-1)/i!^3, i=1..n))
    end:
a:= n-> b(n)*n!^3:
seq(a(n), n=0..20);  # Alois P. Heinz, May 11 2016

b[n_] := b[n] = If[n==0, 1, Sum[b[n-i]*Binomial[n-1, i-1]/i!^3, {i, n}]];
a[n_] := b[n]*n!^3;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 15 2017, after Alois P. Heinz *)

a(n) = Sum_{pi} n!/(k(1)! * 1!^k(1) * k(2)! * 2!^k(2) * ... * k(n)! * n!^k(n)) * (n!/(1!^k(1) * 2!^k(2) * ... * n!^k(n)))^L, where pi runs through all partitions k(1) + 2 * k( 2) + ... + n * k(n) = n, with L = 3.

a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k)^4 * (n-k) * a(k). - Ilya Gutkovskiy, Jul 12 2020

Formula and more terms from Vladeta Jovovic, Dec 09 2001

A061686 Generalized Bell numbers: column 5 of A275043.

Original entry on oeis.org

1, 1, 17, 1540, 461105, 350813126, 573843627152, 1797582928354025, 9904754169831094065, 89944005095677792967482, 1278494002506675052860358142, 27281796399886236251265603339575, 844252087185585895268923657508727440, 36800471170748991972750857754287551544147

Offset: 0

N. J. A. Sloane, Jun 18 2001

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Column k=5 of A275043.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n, k)^5*(n-k)*a(k)/n, k=0..n-1))
    end:
seq(a(n), n=0..15); # Alois P. Heinz, Nov 07 2008

a[n_] := a[n] = If[n == 0, 1, Sum[Binomial[n, k]^5*(n-k)*a[k]/n, {k, 0, n-1}]]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 24 2014, after Alois P. Heinz *)

a61686=[1];A061686(n)={n>1||return(1);#a61686A061686(k))/n} \\ M. F. Hasler, May 11 2015

Sum_{n>=0} a(n) * x^n / (n!)^5 = exp(Sum_{n>=1} x^n / (n!)^5). - Ilya Gutkovskiy, Jul 17 2020

More terms from Alois P. Heinz, Nov 07 2008

A061687 Generalized Bell numbers: column 6 of A275043.

Original entry on oeis.org

1, 1, 33, 8506, 9483041, 33056715626, 293327384637282, 5747475089121405893, 224054040415856117594913, 16044797009828490454609378642, 1981736776623437001042672440089658, 401147408702290404750740714717055504773, 127573929384655691416638350563783440408133922

Offset: 0

N. J. A. Sloane, Jun 18 2001

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..117
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Column k=6 of A275043.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n, k)^6*(n-k)*a(k)/n, k=0..n-1))
    end:
seq(a(n), n=0..15); # Alois P. Heinz, Nov 07 2008

a[n_] := a[n] = If[n == 0, 1, Sum[Binomial[n, k]^6*(n-k)*a[k]/n, {k, 0, n-1}]]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 19 2014, after Alois P. Heinz *)

Sum_{n>=0} a(n) * x^n / (n!)^6 = exp(Sum_{n>=1} x^n / (n!)^6). - Ilya Gutkovskiy, Jul 17 2020

More terms from Alois P. Heinz, Nov 07 2008

A061688 Generalized Bell numbers: column 7 of A275043.

Original entry on oeis.org

1, 1, 65, 48844, 209175233, 3464129078126, 173566857025139312, 22208366234650578141209, 6409515697874502425444186817, 3794729706423816704068204814925754, 4276126299841623727960390049367617509190, 8631647765438316626054238101611711249984175399

Offset: 0

N. J. A. Sloane, Jun 18 2001

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Column k=7 of A275043.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n, k)^7*(n-k)*a(k)/n, k=0..n-1))
    end:
seq(a(n), n=0..12);  # Alois P. Heinz, Nov 07 2008

a[n_] := a[n] = If[n == 0, 1, Sum[Binomial[n, k]^7*(n-k)*a[k]/n, {k, 0, n-1}]]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Apr 17 2014, after Alois P. Heinz *)

Sum_{n>=0} a(n) * x^n / (n!)^7 = exp(Sum_{n>=1} x^n / (n!)^7). - Ilya Gutkovskiy, Jul 17 2020

More terms from Alois P. Heinz, Nov 07 2008

A023998 Number of block permutations on an n-set which are uniform, i.e., corresponding blocks have same size.

Original entry on oeis.org

1, 1, 3, 16, 131, 1496, 22482, 426833, 9934563, 277006192, 9085194458, 345322038293, 15024619744202, 740552967629021, 40984758230303149, 2527342803112928081, 172490568947825135203, 12952575262915522547136, 1064521056888312620947794, 95305764621957309071404877

Offset: 0

Des FitzGerald (D.FitzGerald(AT)utas.edu.au)

Number of games of simple patience with n cards. Take a shuffled deck of n cards labeled 1..n; as each card is dealt it is placed either on a higher-numbered card or starts a new pile to the right. Cards are not moved once they are placed. Suggested by reading Aldous and Diaconis. - N. J. A. Sloane, Dec 19 1999

Number of set partitions of [2n] such that within each block the numbers of odd and even elements are equal. a(2) = 3: 1234, 12|34, 14|23; a(3) = 16: 123456, 1234|56, 1236|45, 1245|36, 1256|34, 12|3456, 12|34|56, 12|36|45, 1346|25, 1456|23, 14|2356, 14|23|56, 16|2345, 16|23|45, 14|25|36, 16|25|34. - Alois P. Heinz, Jul 14 2016

			For n=3 there are 25 block permutations, of which 9 of the form ({1} maps to {1,2}; {2,3} maps to {3}), are not uniform. Hence a(3) = 25 - 9 = 16.
Alternatively, for n=3 the 6 permutations of 3 cards produce 16 games, as follows: 123 -> {1,2,3}; 132 -> {1,32}, {1,3,2}; 213 -> {21,3}, {2,1,3}; 231 -> {21,3}, {2,31}, {2,3,1}; 312 -> {31,2}, {32,1}, {3,1,2}; 321 -> {321}, {32,1}, {31,2}, {3,21}, {3,2,1}.
G.f.: A(x) = 1 + x + 3*x^2/2!^2 + 16*x^3/3!^2 + 131*x^4/4!^2 + 1496*x^5/5!^2 + ...
log(A(x)) = x + x^2/2!^2 + x^3/3!^2 + x^4/4!^2 + x^5/5!^2 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..300
M. Aguiar and R. C. Orellana, The Hopf algebra of uniform block permutations, J. Algebr. Comb. 28 (2008) 115-138
D. Aldous and P. Diaconis, Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. 36 (1999), 413-432.
H. Cheballah, S. Giraudo, and R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.
Fabian Faulstich, Bernd Sturmfels, and Svala Sverrisdóttir, Algebraic Varieties in Quantum Chemistry, arXiv:2308.05258 [math.AG], 2023.
D. G. FitzGerald and Jonathan Leech, Dual symmetric inverse monoids and representation theory, J. Australian Mathematical Society (Series A), Vol. 64 (1998), pp. 345-367.
Raúl E. González-Torres, A geometric study of cores of idempotent stochastic matrices, Linear Algebra Appl. 527, 87-127 (2017).
Rosa Orellana, Franco Saliola, Anne Schilling, and Mike Zabrocki, The lattice of submonoids of the uniform block permutations containing the symmetric group, arXiv:2405.09710 [math.CO], 2024. See p. 3.
Sebastian Volz, Design and Implementation of Efficient Algorithms for Operations on Partitions of Sets, Bachelor Thesis, Saarland Univ. (Germany, 2023). See p. 45.

Cf. A023997, A002720, A061691.
Cf. A132813.
Column k=2 of A275043.
Main diagonal of A321296 and of A322670.

a023998 n = a023998_list !! n
a023998_list = 1 : f 2 [1] a132813_tabl where
   f x ys (zs:zss) = y : f (x + 1) (ys ++ [y]) zss where
                     y = sum $ zipWith (*) ys zs
-- Reinhard Zumkeller, Apr 04 2014

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)*binomial(n-1, i-1)/i!, i=1..n))
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..25);  # Alois P. Heinz, May 11 2016

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[n-1, k] a[k], {k, 0, n-1}];
Array[a, 25, 0] (* Jean-François Alcover, Jul 28 2016 *)
nmax = 20; CoefficientList[Series[E^(-1 + BesselI[0, 2*Sqrt[x]]), {x, 0, nmax}], x]*Range[0, nmax]!^2 (* Vaclav Kotesovec, Jun 09 2019 *)

a(n)=if(n==0,1,sum(k=0,n-1,binomial(n,k)*binomial(n-1,k)*a(k))) \\ Paul D. Hanna, Aug 15 2007

{a(n)=n!^2*polcoeff(exp(sum(m=1, n, x^m/m!^2)+x*O(x^n)), n)} /* Paul D. Hanna */

N=66; x='x+O('x^N); /* that many terms */
Vec(serlaplace(serlaplace(exp(sum(n=1, N, x^n/n!^2))))) /* show terms */
/* Joerg Arndt, Jul 12 2011 */

v=vector(N); v[1]=1;
for (n=1,N-1, v[n+1]=sum(k=0,n-1, binomial(n,k)*binomial(n-1,k)*v[k+1]) );
v /* show terms */
/* Joerg Arndt, Jul 12 2011 */

a(n) = Sum_{k=0..n-1} C(n,k)*C(n-1,k)*a(k) for n>0 with a(0)=1. - Paul D. Hanna, Aug 15 2007
G.f.: Sum_{n>=0} a(n)*x^n/n!^2 = exp( Sum_{n>=1} x^n/n!^2 ). [Paul D. Hanna, Jan 04 2011; merged from duplicate entry A179119]
Row sums of A061691.
Generating function: Let J(z) = Sum_{n>=0} z^n/n!^2. Then exp(J(z)-1) = Sum_{n>=0} a(n)*z^n/n!^2 = 1 + z + 3*z^2/2!^2 + 16*z^3/3!^2 + .... - Peter Bala, Jul 11 2011

More terms from Vladeta Jovovic, Sep 03 2002

A275044 Number of set partitions of [n^2] such that within each block the numbers of elements from all residue classes modulo n are equal for n>0, a(0)=1.

Original entry on oeis.org

1, 1, 3, 64, 25097, 350813126, 293327384637282, 22208366234650578141209, 213426677887357366350726096998529, 344735749788852590196707169431958672823413322, 118966637603805785518622376062965559343297730169187276656138

Offset: 0

Alois P. Heinz, Jul 14 2016

nonn

			a(2) = 3: 1234, 12|34, 14|23.
a(3) = 64: 123456789, 123456|789, 123459|678, 123468|579, ... , 159|267|348, 168|279|345, 189|267|345.

Alois P. Heinz, Table of n, a(n) for n = 0..30
Wikipedia, Partition of a set

Main diagonal of A275043.

b:= proc(n, k) option remember; `if`(k*n=0, 1, add(
       binomial(n, j)^k*(n-j)*b(j, k), j=0..n-1)/n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..12);

b[n_, k_] := b[n, k] = If[k*n == 0, 1, Sum[Binomial[n, j]^k*(n-j)*b[j, k], {j, 0, n-1}]/n];
a[n_] := b[n, n];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, May 27 2018, translated from Maple *)

a(n) = (n!)^n * [x^n] exp(Sum_{k>=1} x^k / (k!)^n). - Ilya Gutkovskiy, Jul 12 2020

A275097 Number of set partitions of [8*n] such that within each block the numbers of elements from all residue classes modulo 8 are equal.

Original entry on oeis.org

1, 1, 129, 286498, 4802367617, 386652630390626, 112344305783644570242, 96703375432646667737903621, 213426677887357366350726096998529, 1081530653290057746718498987187644516546, 11534313393388449518393789691807687515711518754

Offset: 0

Alois P. Heinz, Jul 16 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..91
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Column k=8 of A275043.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n, j)^8*(n-j)*a(j), j=0..n-1)/n)
    end:
seq(a(n), n=0..12);

Sum_{n>=0} a(n) * x^n / (n!)^8 = exp(Sum_{n>=1} x^n / (n!)^8). - Ilya Gutkovskiy, Jul 17 2020

A275098 Number of set partitions of [9*n] such that within each block the numbers of elements from all residue classes modulo 9 are equal.

Original entry on oeis.org

1, 1, 257, 1699300, 112660505345, 44687884101953126, 76502602935955053437072, 451167428778794282789329512425, 7771744024861563765933540267436016385, 344735749788852590196707169431958672823413322, 35650419033178479865362827431736721104304210986866382

Offset: 0

Alois P. Heinz, Jul 16 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..82
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Column k=9 of A275043.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n, j)^9*(n-j)*a(j), j=0..n-1)/n)
    end:
seq(a(n), n=0..12);

Sum_{n>=0} a(n) * x^n / (n!)^9 = exp(Sum_{n>=1} x^n / (n!)^9). - Ilya Gutkovskiy, Jul 17 2020

A275099 Number of set partitions of [10*n] such that within each block the numbers of elements from all residue classes modulo 10 are equal.

Original entry on oeis.org

1, 1, 513, 10136746, 2672797504001, 5260857687009765626, 53531132944198868710856802, 2185249026716732313958375321948613, 297263694975439941710846391262298377605633, 116941828532092016226313310933885429108622288425362

Offset: 0

Alois P. Heinz, Jul 16 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..75
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Column k=10 of A275043.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n, j)^10*(n-j)*a(j), j=0..n-1)/n)
    end:
seq(a(n), n=0..12);

a[n_] := a[n] = If[n==0, 1, Sum[Binomial[n, j]^10*(n-j)*a[j], {j, 0, n-1}]/n];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Jun 27 2022, after Alois P. Heinz *)

Sum_{n>=0} a(n) * x^n / (n!)^10 = exp(Sum_{n>=1} x^n / (n!)^10). - Ilya Gutkovskiy, Jul 17 2020

cp's OEIS Frontend

A061684 Generalized Bell numbers: column 3 of A275043.

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A061685 Generalized Bell numbers: column 4 of A275043.

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A061686 Generalized Bell numbers: column 5 of A275043.

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A061687 Generalized Bell numbers: column 6 of A275043.

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A061688 Generalized Bell numbers: column 7 of A275043.

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A023998 Number of block permutations on an n-set which are uniform, i.e., corresponding blocks have same size.

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A275044 Number of set partitions of [n^2] such that within each block the numbers of elements from all residue classes modulo n are equal for n>0, a(0)=1.

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