A301342 -id:A301342 - cp's OEIS frontend

A301469 Signed recurrence over enriched r-trees: a(n) = 2 * (-1)^n + Sum_y Product_{i in y} a(y) where the sum is over all integer partitions of n - 1.

2, -1, 1, 0, 0, 1, 0, 1, 1, 1, 2, 3, 3, 6, 7, 11, 17, 23, 35, 53, 75, 119, 173, 264, 398, 603, 911, 1411, 2114, 3279, 4977, 7696, 11760, 18253, 27909, 43451, 66675, 103945, 160096, 249904, 385876, 603107, 933474, 1461967, 2266384, 3553167, 5521053, 8664117, 13485744

Offset: 0

Gus Wiseman, Mar 21 2018

sign

Cf. A032305, A055277, A093637, A127524, A196545, A220418, A273866, A273873, A289501, A290261, A290973, A301342-A301345, A301364-A301368, A301422, A301462, A301467, A301470.

a[n_]:=a[n]=2(-1)^n+Sum[Times@@a/@y,{y,IntegerPartitions[n-1]}];
Array[a,30]

O.g.f.: 2/(1 + x) + x Product_{i > 0} 1/(1 - a(i) x^i).

a(n) = Sum_t 2^k * (-1)^w where the sum is over all enriched r-trees of size n, k is the number of leaves, and w is the sum of leaves.

A317580 Number of unlabeled rooted identity trees with n nodes and a distinguished leaf.

Original entry on oeis.org

1, 1, 1, 3, 5, 12, 28, 66, 153, 367, 880, 2121, 5127, 12441, 30248, 73746, 180077, 440571, 1079438, 2648511, 6506170, 16001256, 39393173, 97074140, 239419963, 590972968, 1459808862, 3608483107, 8925476591, 22090139751, 54702648393, 135533335933, 335967782916

Offset: 1

Gus Wiseman, Jul 31 2018

nonn

Total number of leaves in all rooted identity trees with n nodes. - Andrew Howroyd, Aug 28 2018

			The a(6) = 12 rooted identity trees with a distinguished leaf:
(((((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(O))).

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000081, A001678, A003227, A003238, A004111, A038046, A055327, A067824, A301342, A316784.

urit[n_]:=Join@@Table[Select[Union[Sort/@Tuples[urit/@ptn]],UnsameQ@@#&],{ptn,IntegerPartitions[n-1]}];
Table[Sum[Length[Flatten[{t/.{}->1}]],{t,urit[n]}],{n,10}]

WeighMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, (-1)^(i-1)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
seq(n)={my(v=[y]); for(n=2, n, v=concat([y], WeighMT(v))); apply(p -> subst(deriv(p), y, 1), v)} \\ Andrew Howroyd, Aug 28 2018

a(n) = Sum_{k=1, n} k*A055327(n, k). - Andrew Howroyd, Aug 28 2018

Terms a(26) and beyond from Andrew Howroyd, Aug 28 2018

A301365 Regular triangle where T(n,k) is the number of strict trees of weight n with k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 4, 4, 1, 0, 1, 3, 7, 9, 7, 1, 0, 1, 3, 10, 19, 20, 11, 1, 0, 1, 4, 15, 35, 51, 43, 16, 1, 0, 1, 4, 18, 55, 104, 123, 84, 22, 1, 0, 1, 5, 25, 84, 196, 298, 284, 153, 29, 1, 0, 1, 5, 30, 120, 331, 624, 783, 614, 260, 37

Offset: 1

Gus Wiseman, Mar 19 2018

A strict tree of weight n > 0 is either a single node of weight n, or a finite sequence of two or more strict trees with strictly decreasing weights summing to n.

			Triangle begins:
  1
  1   0
  1   1   0
  1   1   1   0
  1   2   2   1   0
  1   2   4   4   1   0
  1   3   7   9   7   1   0
  1   3  10  19  20  11   1   0
  1   4  15  35  51  43  16   1   0
The T(7,3) = 7 strict trees: ((51)1), ((42)1), ((41)2), ((32)2), (4(21)), ((31)3), (421).

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Row sums are A273873.

Cf. A004111, A008284, A032305, A055277, A063834, A281145, A289501, A294018, A294079, A300352, A300442, A300443, A301342, A301364-A301368.

strtrees[n_]:=Prepend[Join@@Table[Tuples[strtrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]>1&&UnsameQ@@#&]}],n];
Table[Length[Select[strtrees[n],Count[#,_Integer,{-1}]===k&]],{n,12},{k,n}]

A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + polcoef(prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)), n)); vector(n, k, Vecrev(v[k]/y, k))}
my(T=A(10));for(n=1, #T, print(T[n])) \\ Andrew Howroyd, Aug 26 2018

cp's OEIS Frontend

A301469 Signed recurrence over enriched r-trees: a(n) = 2 * (-1)^n + Sum_y Product_{i in y} a(y) where the sum is over all integer partitions of n - 1.

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A317580 Number of unlabeled rooted identity trees with n nodes and a distinguished leaf.

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A301365 Regular triangle where T(n,k) is the number of strict trees of weight n with k leaves.

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Author

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Comments

Examples

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Crossrefs

Programs

Mathematica

PARI