A144210
Number of simple graphs on n labeled nodes, where each maximally connected subgraph consists of a single node or has a unique cycle of length 4; also row sums of A144209.
Original entry on oeis.org
1, 1, 1, 1, 4, 76, 1486, 29506, 628531, 14633011, 373486051, 10423892971, 316702467496, 10422938835196, 369779598658786, 14078057663869606, 572776958092098166, 24810200300393961286, 1140218754844983978646
Offset: 0
a(4) = 4, because there are 4 simple graphs on 4 labeled nodes, where each maximally connected subgraph consists of a single node or has a unique cycle of length 4:
.1.2. .1-2. .1-2. .1.2.
..... .|.|. ..X.. .|X|.
.3.4. .3-4. .3-4. .3.4.
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T:= proc(n,k) option remember; if k=0 then 1 elif k<0 or n add(T(n,k), k=0..n): seq(a(n), n=0..23);
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T[n_, k_] := T[n, k] = Which[k == 0, 1, k<0 || nJean-François Alcover, Dec 02 2014, translated from Maple *)
A065889
a(n) = number of unicyclic connected simple graphs whose cycle has length 4.
Original entry on oeis.org
3, 60, 1080, 20580, 430080, 9920232, 252000000, 7015381560, 212840939520, 6998969586180, 248180493969408, 9445533398437500, 384213343210045440, 16639691095281974160, 764619269867445288960, 37163398969133506235952, 1905131520000000000000000
Offset: 4
A065888 ( = 2*
A065889) counts sagittal graphs with one cycle (length 4).
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List([4..25], n-> 12*Binomial(n,4)*n^(n-5)); # G. C. Greubel, May 16 2019
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[12*Binomial(n,4)*n^(n-5) : n in [4..25]]; // G. C. Greubel, May 16 2019
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Table[12*Binomial[n,4]*n^(n-5), {n,4,25}] (* G. C. Greubel, May 16 2019 *)
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{a(n) = 12*binomial(n,4)*n^(n-5)}; \\ G. C. Greubel, May 16 2019
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[12*binomial(n,4)*n^(n-5) for n in (4..25)] # G. C. Greubel, May 16 2019
A144212
Triangle T(n,k), n>=3, 3<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes, where each maximally connected subgraph consists of a single node or has a unique cycle of length k.
Original entry on oeis.org
2, 17, 4, 221, 76, 13, 3261, 1486, 433, 61, 54801, 29506, 11593, 2941, 361, 1049235, 628531, 296353, 102481, 23041, 2521, 22695027, 14633011, 7795873, 3270961, 1010881, 204121, 20161, 548904831, 373486051, 217126225, 104038201, 39355201
Offset: 3
T(4,4) = 4, because there are 4 simple graphs on 4 labeled nodes, where each maximally connected subgraph consists of a single node or has a unique cycle of length 4:
.1.2. .1-2. .1-2. .1.2.
..... .|.|. ..X.. .|X|.
.3.4. .3-4. .3-4. .3.4.
Triangle begins:
2;
17, 4;
221, 76, 13;
3261, 1486, 433, 61;
54801, 29506, 11593, 2941, 361;
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B:= proc(n,c,k) option remember; if c=0 then 1 elif c<0 or n add(B(n,c,k), c=0..n): seq(seq(T(n,k), k=3..n), n=3..11);
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B[n_, c_, k_] := B[n, c, k] = Which[c == 0, 1, c<0 || nJean-François Alcover, Jan 21 2014, translated from Alois P. Heinz's Maple code *)
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