A275100
Number of set partitions of [3*n] 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, 5, 16, 64, 298, 1540, 8506, 48844, 286498, 1699300, 10136746, 60643324, 363328498, 2178376660, 13065476986, 78378513004, 470228031298, 2821239047620, 16927046865226, 101561118929884, 609363226794898, 3656168900416180, 21936982021437466, 131621797985445964
Offset: 0
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CoefficientList[Series[-(21x^3-7x^2-5x+1)/((x-1)(6x-1)(3x-1)),{x,0,30}],x] (* Harvey P. Dale, Dec 15 2018 *)
A275101
Number of set partitions of [4*n] 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, 15, 131, 1613, 25097, 461105, 9483041, 209175233, 4802367617, 112660505345, 2672797504001, 63775070743553, 1526140298561537, 36573850636201985, 877130337148149761, 21043423870122115073, 504949726500343545857, 12117684104978986369025
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..725
- 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.
- Index entries for linear recurrences with constant coefficients, signature (47,-718,4416,-10656,6912).
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LinearRecurrence[{47,-718,4416,-10656,6912},{1,15,131,1613,25097,461105},20] (* Harvey P. Dale, Apr 30 2022 *)
A275102
Number of set partitions of [5*n] 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, 52, 1496, 69026, 4383626, 350813126, 33056715626, 3464129078126, 386652630390626, 44687884101953126, 5260857687009765626, 625229219690048828126, 74663901894300244140626, 8937876284201001220703126, 1071238363160070006103515626, 128470217809820900030517578126
Offset: 0
A345400
Number of colored set partitions of [n] where (i mod d) identifies the color of i and d is the number of available colors such that within each block the frequency of all colors is equal.
Original entry on oeis.org
1, 1, 3, 6, 19, 53, 225, 878, 4281, 21212, 117489, 678571, 4238024, 27644438, 191326221, 1383029112, 10490101937, 82864869805, 682358388107, 5832742205058, 51733248275075, 474870253871245, 4507061060486642, 44152005855084347, 445973953222607799
Offset: 0
a(0) = 1: (), the empty partition.
a(1) = 1: 1a.
a(2) = 3: 1a2b, 1a2a, 1a|2a.
a(3) = 6: 1a2b3c, 1a2a3a, 1a2a|3a, 1a3a|2a, 1a|2a3a, 1a|2a|3a.
a(4) = 19: 1a2b3c4d, 1a2b3a4b, 1a2b|3a4b, 1a4b|2b3a, 1a2a3a4a, 1a2a3a|4a, 1a2a4a|3a, 1a2a|3a4a, 1a2a|3a|4a, 1a3a4a|2a, 1a3a|2a4a, 1a3a|2a|4a, 1a4a|2a3a, 1a|2a3a4a, 1a|2a3a|4a, 1a4a|2a|3a, 1a|2a4a|3a, 1a|2a|3a4a, 1a|2a|3a|4a.
Here the colors a, b, c, ... are used.
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A:= proc(n, k) option remember; `if`(n=0, 1, add(
binomial(n, j)^k*(n-j)*A(j, k), j=0..n-1)/n)
end:
a:= n-> `if`(n=0, 1, add(A(n/d, d), d=numtheory[divisors](n))):
seq(a(n), n=0..28);
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A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Binomial[n, j]^k*(n - j)*A[j, k], {j, 0, n - 1}]/n];
a[n_] := If[n == 0, 1, Sum[A[n/d, d], {d, Divisors[n]}]];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Aug 25 2021, after Alois P. Heinz *)
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