A144524
Triangular numbers n*(n+1)/2 with n composite, where number of prime factors of n, counted with multiplicity, is less than the number of prime factors in n+1.
Original entry on oeis.org
120, 351, 630, 780, 1225, 1326, 1540, 1953, 2016, 2145, 2415, 2775, 3003, 3828, 4186, 4560, 4950, 6216, 6670, 7140, 7626, 7875, 8385, 9045, 10296, 10731, 12090, 12720, 13041, 14365, 15400, 16836, 17205, 17578, 17766, 18915, 19110, 20706, 21321, 21528
Offset: 1
If n = 15 = 2*3 (number of prime factors = 2) and n+1 = 16 = 2*2*2*2 (number of prime factors = 4), then 15*16/2 = 120 = a(1). If n = 26 = 2*13 (number of prime factors = 2) and n+1 = 27 = 3*3*3 (number of prime factors = 3), then 26*27/2 = 351 = a(2). If n = 35 = 5*7 (number of prime factors = 2) and n+1 = 36 = 2*2*3*3 (number of prime factors = 4), then 35*36/2 = 630 = a(3), etc.
A156235
Number of irreflexive binary relations on the power set P(N) of an n-element set N as restricted below.
Original entry on oeis.org
1, 1, 4, 198, 209342
Offset: 0
One of the 209342 irreflexive relations corresponding to a(4) is
R = {({1},{2}), ({2},{1}), ({3,4},{1,2}), ({1,4},{3}), ({2},{3,4})}.
Notice how the last three ordered pairs correspond to jointly giving and/or receiving gifts.
A348451
Triangle read by rows: T(n,k) (1 <= k <= n) is the number of 3-extensions of an n-set over all choices of 3-partitions of the n-set.
Original entry on oeis.org
1, 2, 1, 5, 4, 1, 14, 13, 6, 1, 41, 40, 25, 9, 1, 122, 121, 90, 48, 12, 1, 365, 364, 301, 202, 78, 14, 1, 1094, 1093, 966, 747, 380, 106, 16, 1, 3281, 3280, 3025, 2559, 1571, 592, 141, 18, 1, 9842, 9841, 9330, 8362, 5864, 2755, 906, 180, 20, 1
Offset: 1
Triangle begins:
1,
2,1,
5,4,1,
14,13,6,1,
41,40,25,9,1,
122,121,90,48,12,1,
365,364,301,202,78,14,1,
1094,1093,966,747,380,106,16,1,
3281,3280,3025,2559,1571,592,141,18,1,
9842,9841,9330,8362,5864,2755,906,180,20,1,
...
- Norman Lindquist and Gerard Sierksma, Extensions of set partitions, Journal of Combinatorial Theory, Series A 31.2 (1981): 190-198. See Table III.
A373173
Triangle read by rows: the exponential almost-Riordan array ( exp(exp(x)-1) | exp(x), exp(x)-1 ).
Original entry on oeis.org
1, 1, 1, 2, 1, 1, 5, 1, 3, 1, 15, 1, 7, 6, 1, 52, 1, 15, 25, 10, 1, 203, 1, 31, 90, 65, 15, 1, 877, 1, 63, 301, 350, 140, 21, 1, 4140, 1, 127, 966, 1701, 1050, 266, 28, 1, 21147, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1, 115975, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1
Offset: 0
The triangle begins:
1;
1, 1;
2, 1, 1;
5, 1, 3, 1;
15, 1, 7, 6, 1;
52, 1, 15, 25, 10, 1;
203, 1, 31, 90, 65, 15, 1;
...
-
T[n_,0]:=n!SeriesCoefficient[Exp[Exp[x]-1],{x,0,n}]; T[n_,k_]:=(n-1)!/(k-1)!SeriesCoefficient[Exp[x](Exp[x]-1)^(k-1),{x,0,n-1}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten
Comments