A300351 -id:A300351 - cp's OEIS frontend

A300353 Number of strict trees of weight n with odd leaves.

1, 1, 0, 1, 1, 2, 2, 4, 7, 14, 24, 46, 92, 186, 368, 750, 1529, 3160, 6510, 13590, 28374, 59780, 125732, 266468, 564188, 1202842, 2560106, 5484304, 11732400, 25229068, 54187918, 116938702, 252039411, 545593378, 1179545874, 2560009400, 5550315640, 12075064432

Offset: 0

Gus Wiseman, Mar 03 2018

nonn

This sequence is initially dominated by A300352 but eventually becomes much greater.

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

			The a(8) = 7 strict trees with odd leaves: (71), (53), (((51)1)1), (((31)3)1), (((31)1)3), ((31)31), (((((31)1)1)1)1).

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A000009, A063834, A078408, A089259, A196545, A279374, A279785, A289501, A294018, A294079, A299203, A300300, A300301, A300351, A300352, A300354, A300355.

d[n_]:=d[n]=If[EvenQ[n],0,1]+Sum[Times@@d/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&UnsameQ@@#&]}];
Table[d[n],{n,40}]

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n] = polcoef(x/(1-x^2) + prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)), n)); concat([1], v)} \\ Andrew Howroyd, Aug 25 2018

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

a(n) = Sum_{i=1..A000009(n)} A294018(A300351(n,i)).

A300355 Number of enriched p-trees of weight n with odd leaves.

Original entry on oeis.org

1, 1, 1, 3, 6, 16, 47, 132, 410, 1254, 4052, 12818, 42783, 139082, 469924, 1563606, 5353966, 18065348, 62491018, 213391790, 743836996, 2565135934, 8994087070, 31251762932, 110245063771, 385443583008, 1365151504722, 4800376128986, 17070221456536, 60289267885410

Offset: 0

Gus Wiseman, Mar 03 2018

nonn

An enriched p-tree of weight n > 0 is either a single node of weight n, or a sequence of two or more enriched p-trees with weakly decreasing weights summing to n.

			The a(5) = 16 enriched p-trees of weight with odd leaves:
5,
((31)1), ((((11)1)1)1), (((111)1)1), (((11)(11))1), (((11)11)1), ((1111)1),
(3(11)), (((11)1)(11)), ((111)(11)),
(311), (((11)1)11), ((111)11),
((11)(11)1),
((11)111),
(11111).

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A000009, A063834, A078408, A089259, A196545, A279374, A279785, A289501, A294079, A299202, A299203, A300300, A300301, A300352, A300353, A300354.

c[n_]:=c[n]=If[EvenQ[n],0,1]+Sum[Times@@c/@y,{y,Select[IntegerPartitions[n],Length[#]>1&]}];
Table[c[n],{n,30}]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = n%2 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)); concat([1], v)} \\ Andrew Howroyd, Aug 26 2018

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

a(n) = Sum_{i=1..A000009(n)} A299203(A300351(n,i)).

cp's OEIS Frontend

A300353 Number of strict trees of weight n with odd leaves.

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