A360335
Array read by antidiagonals downwards: A(n,m) = number of set partitions of [2n] into 2-element subsets {i, i+k} with 1 <= k <= m.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 7, 5, 1, 1, 3, 12, 16, 8, 1, 1, 3, 15, 35, 38, 13, 1, 1, 3, 15, 63, 105, 89, 21, 1, 1, 3, 15, 90, 226, 329, 209, 34, 1, 1, 3, 15, 105, 417, 841, 1014, 491, 55, 1, 1, 3, 15, 105, 645, 1787, 3251, 3116, 1153, 89, 1
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 3, 3, 3, 3, 3, 3, ...
1, 3, 7, 12, 15, 15, 15, 15, 15, ...
1, 5, 16, 35, 63, 90, 105, 105, 105, ...
1, 8, 38, 105, 226, 417, 645, 840, 945, ...
1, 13, 89, 329, 841, 1787, 3348, 5445, 7665, ...
1, 21, 209, 1014, 3251, 7938, 16717, 31647, 53250, ...
1, 34, 491, 3116, 12483, 36500, 86311, 180560, 344403, ...
1, 55, 1153, 9610, 47481, 167631, 459803, 1062435, 2211181, ...
...
A320346
a(n) is the number of perfect matchings in the graph with vertices labeled 1 to 2n with edges {i,j} for 1<=|i-j|<=4.
Original entry on oeis.org
1, 1, 3, 12, 35, 105, 329, 1014, 3116, 9610, 29625, 91279, 281303, 866948, 2671727, 8233671, 25374513, 78198928, 240992592, 742688720, 2288811009, 7053635369, 21737825143, 66991419284, 206453506615, 636246416105, 1960778041673, 6042706771910, 18622355183932, 57390193784986, 176864543185497
Offset: 0
The a(3) = 12 matchings are (12)(34)(56), (12)(35)(46), (12)(36)(45), (13)(24)(56), (13)(25)(46), (13)(26)(45), (14)(23)(56), (14)(25)(36), (14)(26)(35), (15)(23)(46), (15)(24)(36), (15)(26)(34).
- Robert Israel, Table of n, a(n) for n = 0..2044
- M. Schwartz, Efficiently computing the permanent and Hafnian of some banded Toeplitz matrices, Linear Algebra and its Applications 430 (2009), 1364-1374.
- Index entries for linear recurrences with constant coefficients, signature (2,1,6,3,2,1,-2,-1).
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f:= gfun:-rectoproc({a(n) + 2*a(n + 1) - a(n + 2) - 2*a(n + 3) - 3*a(n + 4) - 6*a(n + 5)- a(n + 6) - 2*a(n + 7) + a(n + 8), a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 12, a(4) = 35, a(5) = 105, a(6) = 329, a(7) = 1014}, a(n), remember):
map(f, [$0..100]);
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LinearRecurrence[{2, 1, 6, 3, 2, 1, -2, -1}, {1, 1, 3, 12, 35, 105, 329, 1014}, 40] (* Jean-François Alcover, Apr 30 2019 *)
Showing 1-2 of 2 results.