A241500
Triangle T(n,k): number of ways of partitioning the n-element multiset {1,1,2,3,...,n-1} into exactly k nonempty parts, n>=1 and 1<=k<=n.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 5, 4, 1, 1, 11, 16, 7, 1, 1, 23, 58, 41, 11, 1, 1, 47, 196, 215, 90, 16, 1, 1, 95, 634, 1041, 640, 176, 22, 1, 1, 191, 1996, 4767, 4151, 1631, 315, 29, 1, 1, 383, 6178, 21001, 25221, 13587, 3696, 526, 37, 1, 1, 767, 18916, 90055, 146140, 105042, 38409, 7638, 831, 46, 1
Offset: 1
There are 58 ways to partition {1,1,2,3,4,5} into three nonempty parts.
The first few rows are:
1;
1, 1;
1, 2, 1;
1, 5, 4, 1;
1, 11, 16, 7, 1;
1, 23, 58, 41, 11, 1;
1, 47, 196, 215, 90, 16, 1;
1, 95, 634, 1041, 640, 176, 22, 1;
1, 191, 1996, 4767, 4151, 1631, 315, 29, 1;
1, 383, 6178, 21001, 25221, 13587, 3696, 526, 37, 1;
...
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T(n,k) = stirling(n-1,k,2) + stirling(n-1,k-1,2) + binomial(k,2)*stirling(n-2,k,2); \\ Michel Marcus, Apr 24 2014
A168584
Number of ways of partitioning the multiset {1,1,2,3,...,n-1} into exactly four nonempty parts.
Original entry on oeis.org
1, 7, 41, 215, 1041, 4767, 21001, 90055, 378881, 1572527, 6463161, 26375895, 107081521, 433076287, 1746588521, 7029269735, 28245956961, 113370724047, 454644109081, 1822061123575, 7298700653201, 29226175283807
Offset: 4
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[(5*4^(n-3)-3^(n-1)+3*2^(n-2)-2)/3: n in [4..30]]; // Wesley Ivan Hurt, Dec 12 2015
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A168584:=n->(5*4^(n-3)-3^(n-1)+3*2^(n-2)-2)/3: seq(A168584(n), n=4..30); # Wesley Ivan Hurt, Dec 12 2015
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f2[n_] := 1/3 (5 4^(n - 3) - 3^(n - 1) + 3 2^(n - 2) - 2); Table[f2[n], {n, 4, 25}]
LinearRecurrence[{10,-35,50,-24},{1,7,41,215},30] (* Harvey P. Dale, Sep 15 2020 *)
A168585
Number of ways of partitioning the multiset {1,1,2,3,...,n-1} into exactly five nonempty parts.
Original entry on oeis.org
1, 11, 90, 640, 4151, 25221, 146140, 817730, 4458201, 23835031, 125551790, 653873220, 3375658651, 17308994441, 88284419040, 448429907110, 2270331053501, 11464832543451, 57778226219890, 290711449879400
Offset: 5
The number of ways of partitioning the multiset {1, 1, 2, 3, ..., n-1} into exactly two, three and four nonempty parts are given in
A083329,
A168583 and
A168584, respectively.
A158198
Triangle T(n, k) = Sum_{j=0..k-1} (-1)^j*binomial(k, j+1)*(k-j)^(n-k), read by rows.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 7, 4, 1, 1, 15, 16, 5, 1, 1, 31, 58, 25, 6, 1, 1, 63, 196, 125, 36, 7, 1, 1, 127, 634, 601, 216, 49, 8, 1, 1, 255, 1996, 2765, 1296, 343, 64, 9, 1, 1, 511, 6178, 12265, 7656, 2401, 512, 81, 10, 1, 1, 1023, 18916, 52925, 44136, 16807, 4096, 729, 100, 11, 1
Offset: 1
Thomas J Engelsma (tom(AT)opertech.com), Mar 13 2009
Triangle begins
1;
1, 1;
1, 3, 1;
1, 7, 4, 1;
1, 15, 16, 5, 1;
1, 31, 58, 25, 6, 1;
1, 63, 196, 125, 36, 7, 1;
1, 127, 634, 601, 216, 49, 8, 1;
1, 255, 1996, 2765, 1296, 343, 64, 9, 1;
a(8,4) = 1*4*4^4 - 1*6*3^4 + 1*4*2^4 - 1*1*1^4 = 1024 - 486 + 64 - 1 = 601.
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[(&+[(-1)^j*Binomial(k, j+1)*(k-j)^(n-k): j in [0..k-1]]): k in [1..n], n in [1..12]]; // G. C. Greubel, Jun 26 2021
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Table[Sum[(-1)^j*Binomial[k, j+1]*(k-j)^(n-k), {j, 0, k-1}], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jun 26 2021 *)_
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tabl(nn) = {for (n=1, nn, for (k=1, n, print1(sum(i=0, k-1, (-1)^i*binomial(k,i+1)*(k-i)^(n-k)), ", ");); print(););} \\ Michel Marcus, Jun 19 2013
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flatten([[ sum( (-1)^j*binomial(k, j+1)*(k-j)^(n-k) for j in (0..k-1)) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jun 26 2021
Showing 1-4 of 4 results.
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