A048816
Number of rooted trees with n nodes with every leaf at the same height.
Original entry on oeis.org
1, 1, 2, 3, 5, 7, 12, 17, 28, 42, 68, 103, 168, 260, 420, 665, 1075, 1716, 2787, 4489, 7304, 11849, 19333, 31504, 51561, 84347, 138378, 227096, 373445, 614441, 1012583, 1669774, 2756951, 4555183, 7533988, 12469301, 20655523, 34238310, 56795325, 94270949
Offset: 1
See Arndt link.
From _Gus Wiseman_, Oct 08 2018: (Start)
The a(1) = 1 through a(7) = 12 balanced rooted trees with n nodes:
o (o) (oo) (ooo) (oooo) (ooooo) (oooooo)
((o)) ((oo)) ((ooo)) ((oooo)) ((ooooo))
(((o))) (((oo))) (((ooo))) (((oooo)))
((o)(o)) ((o)(oo)) ((o)(ooo))
((((o)))) ((((oo)))) ((oo)(oo))
(((o)(o))) ((((ooo))))
(((((o))))) (((o)(oo)))
((o)(o)(o))
(((((oo)))))
((((o)(o))))
(((o))((o)))
((((((o))))))
(End)
-
T[n_, k_] := T[n, k] = If[n==1, 1, If[k==0, 0, Sum[Sum[If[dJean-François Alcover, Jan 08 2016, after Alois P. Heinz *)
A244925
Number T(n,k) of n-node unlabeled rooted trees with every leaf at height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 4, 3, 2, 1, 1, 0, 1, 4, 5, 3, 2, 1, 1, 0, 1, 7, 7, 6, 3, 2, 1, 1, 0, 1, 8, 12, 8, 6, 3, 2, 1, 1, 0, 1, 12, 18, 15, 9, 6, 3, 2, 1, 1, 0, 1, 14, 27, 23, 16, 9, 6, 3, 2, 1, 1, 0, 1, 21, 42, 39, 26, 17, 9, 6, 3, 2, 1, 1
Offset: 1
The A048816(5) = 5 rooted trees with 5 nodes with every leaf at the same height sorted by height are:
: o : o o : o : o :
: /( )\ : / \ | : | : | :
: o o o o : o o o : o : o :
: : | | /|\ : | : | :
: : o o o o o : o : o :
: : : / \ : | :
: : : o o : o :
: : : : | :
: : : : o :
: : : : :
: ---1--- : -----2----- : --3-- : -4- :
Thus row 5 = [0, 1, 2, 1, 1].
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 1, 1;
0, 1, 2, 1, 1;
0, 1, 2, 2, 1, 1;
0, 1, 4, 3, 2, 1, 1;
0, 1, 4, 5, 3, 2, 1, 1;
0, 1, 7, 7, 6, 3, 2, 1, 1;
0, 1, 8, 12, 8, 6, 3, 2, 1, 1;
0, 1, 12, 18, 15, 9, 6, 3, 2, 1, 1;
0, 1, 14, 27, 23, 16, 9, 6, 3, 2, 1, 1;
...
Columns k=0-10 give:
A000007(n-1),
A000012 (for n>0),
A002865(n-1) (for n>2),
A048808,
A048809,
A048810,
A048811,
A048812,
A048813,
A048814,
A048815.
-
with(numtheory):
T:= proc(n, k) option remember; `if`(n=1, 1, `if`(k=0, 0,
add(add(`if`(d
-
T[n_, k_] := T[n, k] = If[n == 1, 1, If[k == 0, 0, Sum[ Sum[ If[dJean-François Alcover, Jan 28 2015, after Alois P. Heinz *)
A048809
Number of rooted trees with n nodes with every leaf at height 4.
Original entry on oeis.org
1, 1, 2, 3, 6, 8, 15, 23, 39, 61, 102, 161, 265, 420, 682, 1087, 1753, 2790, 4476, 7120, 11376, 18075, 28785, 45666, 72530, 114882, 182040, 287878, 455231, 718755, 1134491, 1788461, 2818140, 4436000, 6978932, 10969695, 17232572, 27049320
Offset: 5
A048815
Number of rooted trees with n nodes with every leaf at height 10.
Original entry on oeis.org
1, 1, 2, 3, 6, 9, 17, 28, 49, 83, 145, 245, 425, 724, 1246, 2130, 3659, 6254, 10724, 18335, 31396, 53676, 91832, 156944, 268324, 458435, 783324, 1337862, 2284950, 3901211, 6660304, 11367935, 19401130, 33104598, 56481086, 96349147
Offset: 11
A048810
Number of rooted trees with n nodes with every leaf at height 5.
Original entry on oeis.org
1, 1, 2, 3, 6, 9, 16, 26, 44, 73, 123, 203, 340, 563, 935, 1550, 2571, 4251, 7034, 11618, 19188, 31654, 52201, 85999, 141631, 233074, 383375, 630215, 1035508, 1700501, 2791309, 4579587, 7510280, 12310980, 20172075, 33039130, 54092556
Offset: 6
A048811
Number of rooted trees with n nodes with every leaf at height 6.
Original entry on oeis.org
1, 1, 2, 3, 6, 9, 17, 27, 47, 78, 135, 224, 384, 642, 1088, 1827, 3088, 5182, 8736, 14661, 24660, 41378, 69500, 116534, 195509, 327627, 549104, 919593, 1539985, 2577399, 4313102, 7214374, 12064930, 20169283, 33710370, 56324729, 94089240
Offset: 7
A048812
Number of rooted trees with n nodes with every leaf at height 7.
Original entry on oeis.org
1, 1, 2, 3, 6, 9, 17, 28, 48, 81, 140, 236, 405, 686, 1169, 1984, 3375, 5723, 9721, 16478, 27949, 47354, 80245, 135869, 230054, 389304, 658706, 1114072, 1883900, 3184602, 5382321, 9094154, 15362767, 25946131, 43811971, 73964065
Offset: 8
A048813
Number of rooted trees with n nodes with every leaf at height 8.
Original entry on oeis.org
1, 1, 2, 3, 6, 9, 17, 28, 49, 82, 143, 241, 417, 707, 1213, 2065, 3534, 6014, 10272, 17487, 29820, 50758, 86469, 147123, 250429, 425932, 724517, 1231765, 2094116, 3558799, 6047447, 10273349, 17450221, 29633832, 50317376, 85420630
Offset: 9
A048814
Number of rooted trees with n nodes with every leaf at height 9.
Original entry on oeis.org
1, 1, 2, 3, 6, 9, 17, 28, 49, 83, 144, 244, 422, 719, 1234, 2109, 3615, 6173, 10565, 18042, 30839, 52653, 89927, 153462, 261931, 446818, 762190, 1299678, 2215990, 3777230, 6437673, 10969447, 18688879, 31834676, 54220089, 92331502
Offset: 10
A074045
Number of rooted trees of 2n+1 nodes with every leaf at height n.
Original entry on oeis.org
1, 1, 2, 3, 6, 9, 17, 28, 49, 83, 145, 246, 427, 729, 1256, 2152, 3702, 6341, 10892, 18662, 32016, 54853, 94034, 161055, 275929, 472461, 809033, 1384848, 2370434, 4056309, 6940744, 11873769, 20311018, 34737781, 59405959, 101577454, 173669088, 296890268
Offset: 0
Showing 1-10 of 10 results.
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