A056861
Triangle T(n,k) is the number of restricted growth strings (RGS) of set partitions of {1..n} that have an increase at index k (1<=k
1, 3, 2, 10, 7, 6, 37, 27, 23, 21, 151, 114, 97, 88, 83, 674, 523, 446, 403, 378, 363, 3263, 2589, 2217, 1999, 1867, 1785, 1733, 17007, 13744, 11829, 10658, 9923, 9452, 9145, 8942, 94828, 77821, 67340, 60689, 56380, 53541, 51644, 50361, 49484, 562595
Offset: 2
A056862
Triangle T(n,k) is the number of restricted growth strings (RGS) of set partitions of {1..n} that have a decrease at index k (1<=k
0, 0, 1, 0, 3, 4, 0, 10, 14, 16, 0, 37, 54, 63, 68, 0, 151, 228, 271, 296, 311, 0, 674, 1046, 1264, 1396, 1478, 1530, 0, 3263, 5178, 6349, 7084, 7555, 7862, 8065, 0, 17007, 27488, 34139, 38448, 41287, 43184, 44467, 45344, 0, 94828, 155642, 195494, 222044, 239976, 252230, 260690, 266584, 270724
Offset: 2
Comments
Number of falls s_k > s_{k+1} in a RGS [s_1, ..., s_n] of a set partition of {1, ..., n}, where s_i is the subset containing i, s_1 = 1 and s_i <= 1 + max(j
Note that the number of equalities at any index is B(n-1), where B(n) are the Bell numbers. - Franklin T. Adams-Watters, Jun 08 2006
Examples
For example, [1, 2, 1, 2, 2, 3] is the RGS of a set partition of {1, 2, 3, 4, 5, 6} and has 1 fall, at i = 2.
0;
0,1;
0,3,4;
0,10,14,16;
0,37,54,63,68;
0,151,228,271,296,311;
0,674,1046,1264,1396,1478,1530;
0,3263,5178,6349,7084,7555,7862,8065;
0,17007,27488,34139,38448,41287,43184,44467,45344;
0,94828,155642,195494,222044,239976,252230,260690,266584,270724;
0,562595,935534,1186845,1358452,1476959,1559602,1617737,1658952,1688379, 1709526;
References
- W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000. [apparently unpublished, Joerg Arndt, Mar 05 2016]
Links
- Alois P. Heinz, Rows n = 2..100, flattened
Programs
-
Maple
b:= proc(n, i, m, t) option remember; `if`(n=0, [1, 0], add((p-> p+[0, `if`(j -
Mathematica
b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, {1, 0}, Sum[Function[p, p + {0, If[j
Formula
T(n,k) = B(n) - B(n-1) - A056861(n,k). - Franklin T. Adams-Watters, Jun 08 2006
Conjecture: T(n,3) = 2*A011965(n). - R. J. Mathar, Mar 08 2016
Extensions
Edited and extended by Franklin T. Adams-Watters, Jun 08 2006
Clarified definition and edited comment and example, Joerg Arndt, Mar 08 2016
Data corrected, R. J. Mathar, Mar 08 2016
A154108 A000110 / (1,2,3,...): (convolved with (1,2,3,...) = Bell numbers).
1, 0, 2, 7, 27, 114, 523, 2589, 13744, 77821, 467767, 2972432, 19895813, 139824045, 1028804338, 7905124379, 63287544055, 526827208698, 4551453462543, 40740750631417, 377254241891064, 3608700264369193, 35613444194346451, 362161573323083920, 3790824599495473121
Offset: 1
Keywords
Comments
This is the sequence which must be convolved with (1,2,3,...), offset 0, to generate the Bell numbers starting (1, 2, 5, 15, 52, ...) offset 1;
equivalent to row sums of triangle A154109 = (1, 2, 5, 15, 52, ...).
A variant of A011965. - R. J. Mathar, Jan 07 2009
Examples
A000110(5) = 52 = (1, 0, 2, 7, 27) convolved with (1, 2, 3, 4, 5) = (5 + 0 + 6 + 14 + 27).
Programs
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Mathematica
nmax = 30; Table[a[j]/.SolveAlways[Table[Sum[a[k]*(n-k), {k, 0, n}]==BellB[n], {n, 1, nmax+1}], a][[1]], {j, 0, nmax}] (* Vaclav Kotesovec, Jul 26 2021 *)
Formula
A000110 / (1,2,3,...); where A000110 (the Bell numbers) begins with offset 1: (1, 2, 5, 15, 52, 203, 877, ...).
G.f.: (A000110(x)-1)*(x-1)^2, where A000110(x) is the g.f. of the Bell numbers. - R. J. Mathar, Nov 27 2018
Extensions
More terms from Vaclav Kotesovec, Jul 26 2021
A191099 5th differences of Bell numbers.
11, 52, 255, 1335, 7432, 43833, 272947, 1788850, 12303997, 88586135, 666047210, 5218287687, 42518759887, 359651145332, 3152929344235, 28603584325827, 268159523175744, 2594608337127709, 25878365376280647, 265770087291261082, 2807571511844891521
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
- Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
Programs
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Mathematica
Differences[BellB[Range[0, 50]], 5]
Comments
Examples
For example, [1, 2, 1, 2, 2, 3] is the RGS of a set partition of {1, 2, 3, 4, 5, 6} and has 3 rises, at i = 1, i = 3 and i = 5. 1; 3,2; 10,7,6; 37,27,23,21; 151,114,97,88,83; 674,523,446,403,378,363; 3263,2589,2217,1999,1867,1785,1733; 17007,13744,11829,10658,9923,9452,9145,8942; 94828,77821,67340,60689,56380,53541,51644,50361,49484; 562595,467767,406953,367101,340551,322619,310365,301905,296011,291871; 3535027,2972432,2599493,2348182,2176575,2058068,1975425,1917290,1876075, 1846648,1825501;References
Links
Crossrefs
Programs
Mathematica
b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, {1, 0}, Sum[Function[p, p + {0, If[jJean-François Alcover, May 23 2016, after Alois P. Heinz *) Extensions