A059584
Triangle T(n,m) of number of labeled m-node T_0-hypergraphs with n hyperedges (empty hyperedges and multiple hyperedges included), m=0,1,...,2^n.
Original entry on oeis.org
1, 1, 1, 2, 2, 1, 3, 7, 12, 12, 1, 4, 16, 68, 292, 1120, 3360, 6720, 6720, 1, 5, 30, 235, 2251, 23520, 245280, 2412480, 21631680, 172972800, 1210809600, 7264857600, 36324288000, 145297152000, 435891456000, 871782912000, 871782912000
Offset: 0
Triangle starts:
1, 1;
1, 2, 2;
1, 3, 7, 12, 12;
1, 4, 16, 68, 292, 1120, 3360, 6720, 6720;
...
There are 7 2-node T_0-hypergraphs with 2 hyperedges: {{}, {1}}, {{}, {2}}, {{1}, {1}}, {{1}, {2}}, {{1}, {1, 2}}, {{2}, {2}} and {{2}, {1, 2}}.
A059585
Number of labeled 3-node T_0-hypergraphs with n hyperedges (empty hyperedges and multiple hyperedges included).
Original entry on oeis.org
0, 0, 12, 68, 235, 636, 1478, 3088, 5958, 10800, 18612, 30756, 49049, 75868, 114270, 168128, 242284, 342720, 476748, 653220, 882759, 1178012, 1553926, 2028048, 2620850, 3356080, 4261140, 5367492, 6711093, 8332860, 10279166, 12602368
Offset: 0
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
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[n*(n-1)*(n+1)*(n^4+28*n^3+323*n^2+1988*n+ 4572)/5040: n in [0..35]]; // Vincenzo Librandi, Oct 07 2017
-
for n from 0 to 100 do printf(`%d,`,n*(n - 1)*(n + 1)*(n^4 + 28*n^3 + 323*n^2 + 1988*n + 4572)/5040) od:
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CoefficientList[Series[x^2*(2 - x)^2*(3 - 4*x + 2*x^2)/(1 - x)^8, {x, 0, 50}], x] (* G. C. Greubel, Oct 06 2017 *)
LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 0, 12, 68, 235, 636, 1478, 3088}, 33] (* Vincenzo Librandi, Oct 07 2017 *)
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x='x+O('x^50); concat([0,0],Vec(x^2*(2-x)^2*(3-4*x+2*x^2)/(1-x)^8)) \\ G. C. Greubel, Oct 06 2017
A059587
T(n,m) = (1/m!)*Sum_{i=0..m} stirling1(m,i)*(2^i)*(2^i+1)*...*(2^i+n-1).
Original entry on oeis.org
1, 1, 1, 2, 1, 2, 6, 7, 4, 1, 6, 24, 48, 68, 73, 56, 28, 8, 1, 24, 120, 360, 940, 2251, 4704, 8176, 11488, 12876, 11440, 8008, 4368, 1820, 560, 120, 16, 1, 120, 720, 3000, 12720, 56660, 247016, 987252, 3480536, 10647035, 28163200, 64592320, 129068160
Offset: 0
Triangle starts:
1, 1;
1, 2, 1;
2, 6, 7, 4, 1;
6, 24, 48, 68, 73, 56, 28, 8, 1;
...
-
with(combinat): for n from 0 to 10 do for m from 0 to 2^n do printf(`%d,`,sum(abs(stirling1(n,i))*binomial(2^i, m), i=0..n)) od: od:
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