A300355 -id:A300355 - cp's OEIS frontend

A300436 Number of odd p-trees of weight n (all proper terminal subtrees have odd weight).

1, 1, 1, 2, 2, 5, 5, 12, 13, 35, 37, 98, 107, 304, 336, 927, 1037, 3010, 3367, 9585, 10924, 32126, 36438, 105589, 121045, 359691, 412789, 1211214, 1398168, 4188930, 4831708, 14315544, 16636297, 50079792, 58084208, 173370663, 202101971, 611487744, 712709423

Offset: 1

Gus Wiseman, Mar 05 2018

nonn

An odd p-tree of weight n > 0 is either a single node (if n = 1) or a finite sequence of at least 3 odd p-trees whose weights are weakly decreasing odd numbers summing to n.

			The a(7) = 5 odd p-trees: ((ooo)(ooo)o), (((ooo)oo)oo), ((ooooo)oo), ((ooo)oooo), (ooooooo).

Cf. A000009, A027193, A063834, A078408, A196545, A279374, A279785, A289501, A298118, A299202, A299203, A300300, A300301, A300355, A300439, A300440.

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

O.g.f: x + Product_{n odd} 1/(1 - a(n)*x^n) - Sum_{n odd} a(n)*x^n. - Gus Wiseman, Aug 27 2018

Name corrected by Gus Wiseman, Aug 27 2018

A300352 Number of strict trees of weight n with distinct leaves.

Original entry on oeis.org

1, 1, 2, 2, 3, 6, 8, 11, 17, 40, 48, 76, 109, 159, 400, 470, 745, 1057, 1576, 2103, 5267, 6022, 9746, 13390, 20099, 26542, 39396, 82074, 101387, 152291, 215676, 308937, 423587, 596511, 799022, 1623311, 1960223, 2947722, 4048704, 5845982, 7794809, 11028888

Offset: 1

Gus Wiseman, Mar 03 2018

nonn

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) = 11 strict trees with distinct leaves: 8, (71), ((52)1), ((43)1), (62), ((51)2), (53), ((41)3), (5(21)), (521), (431).

Cf. A000009, A056239, A063834, A196545, A246867, A273873, A281145, A289501, A294018, A294079, A296150, A299201, A299202, A299203, A300353, A300354, A300355.

sps[{}]:={{}};sps[set:{i_,_}]:=
Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
str[q_]:=str[q]=If[Length[q]===1,1,Total[Times@@@Map[str,Select[sps[q],And[Length[#]>1,UnsameQ@@Total/@#]&],{2}]]];
Table[Total[str/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,1,20}]

a(n) = Sum_{i=1..A000009(n)} A294018(A246867(n,i)).

A292137 G.f.: Im(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, -1, -2, -2, -2, -2, -3, -3, -2, -2, -2, -1, 1, 2, 2, 4, 6, 7, 8, 10, 13, 14, 14, 15, 17, 17, 15, 15, 16, 14, 10, 8, 6, 1, -5, -10, -14, -21, -31, -38, -43, -53, -64, -71, -77, -86, -97, -104, -108, -115, -124, -127, -125, -127, -130, -125, -116

Offset: 0

Seiichi Manyama, Sep 09 2017

			Product_{k>=1} 1/(1 - i*x^k) = 1 + (0+1i)*x + (-1+1i)*x^2 + (-1+0i)*x^3 + (-1+0i)*x^4 + (-1+0i)*x^5 + (-2-1i)*x^6 + (-1-2i)*x^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A000108, A010815, A027193, A063834, A067659, A081362, A099323, A196545, A220418, A290261, A292042, A292043, A292136, A292138, A298118, A300355.

N:= 100:
S := convert(series( add( (-1)^n*x^(2*n+1)/(mul(1 - x^k,k = 1..2*n+1)), n = 0..N ), x, N+1 ), polynom):
seq(coeff(S, x, n), n = 0..N); # Peter Bala, Jan 15 2021

Im[CoefficientList[Series[1/QPochhammer[I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 17 2017 *)

1/(i*x; x)_inf is the g.f. for A292136(n) + i*a(n).
a(n) = Sum (-1)^((k - 1)/2) where the sum is over all integer partitions of n into an odd number of parts and k is the number of parts. - Gus Wiseman, Mar 08 2018
G.f.: Sum_{n >= 0} (-1)^n * x^(2*n+1)/Product_{k = 1..2*n+1} (1 - x^k). - Peter Bala, Jan 15 2021

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

Original entry on oeis.org

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)).

A300354 Number of enriched p-trees of weight n with distinct leaves.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 8, 8, 13, 17, 54, 56, 98, 125, 195, 500, 606, 921, 1317, 1912, 2635, 6667, 7704, 12142, 16958, 24891, 33388, 47792, 106494, 126475, 195475, 268736, 393179, 523775, 750251, 979518, 2090669, 2457315, 3759380, 5066524, 7420874, 9726501, 13935546

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(6) = 8 enriched p-trees with distinct leaves: 6, (42), (51), ((31)2), ((32)1), (3(21)), ((21)3), (321).

Cf. A000009, A000041, A063834, A196545, A246867, A273873, A281145, A289501, A290261, A294018, A296150, A299201, A299202, A299203, A300352, A300353, A300355.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
ept[q_]:=ept[q]=If[Length[q]===1,1,Total[Times@@@Map[ept,Join@@Function[sptn,Join@@@Tuples[Permutations/@GatherBy[sptn,Total]]]/@Select[sps[q],Length[#]>1&],{2}]]];
Table[Total[ept/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,1,30}]

a(n) = Sum_{i=1..A000009(n)} A299203(A246867(n,i)).

A318485 Number of p-trees of weight 2n + 1 in which all outdegrees are odd.

Original entry on oeis.org

1, 1, 2, 5, 13, 37, 107, 336, 1037, 3367, 10924, 36438, 121045, 412789, 1398168, 4831708, 16636297, 58084208, 202101971, 712709423, 2502000811, 8880033929, 31428410158, 112199775788, 399383181020, 1433385148187, 5128572792587, 18481258241133

Offset: 0

Gus Wiseman, Aug 27 2018

nonn

A p-tree of weight n with odd outdegrees is either a single node (if n = 1) or a finite odd-length sequence of at least 3 p-trees with odd outdegrees whose weights are weakly decreasing and sum to n.

			The a(4) = 13 p-trees of weight 9 with odd outdegrees:
  ((((ooo)oo)oo)oo)
  (((ooo)(ooo)o)oo)
  (((ooo)oo)(ooo)o)
  ((ooo)(ooo)(ooo))
  (((ooooo)oo)oo)
  (((ooo)oooo)oo)
  ((ooooo)(ooo)o)
  (((ooo)oo)oooo)
  ((ooo)(ooo)ooo)
  ((ooooooo)oo)
  ((ooooo)oooo)
  ((ooo)oooooo)
  (ooooooooo)

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

Cf. A027193, A063834, A078408, A196545, A279374, A289501, A298118, A300300, A300301, A300355, A300436, A300647, A300652, A300797, A302243.

b[n_]:=b[n]=If[n>1,0,1]+Sum[Times@@b/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&]}];
Table[b[n],{n,1,20,2}]

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n] = polcoef(1/prod(k=1, n-1, 1 - v[k]*x^(2*k-1) + O(x^(2*n))) - 1/prod(k=1, n-1, 1 + v[k]*x^(2*k-1) + O(x^(2*n))), 2*n-1)/2); v} \\ Andrew Howroyd, Aug 27 2018

cp's OEIS Frontend

A300436 Number of odd p-trees of weight n (all proper terminal subtrees have odd weight).

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A300352 Number of strict trees of weight n with distinct leaves.

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A292137 G.f.: Im(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

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

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A300354 Number of enriched p-trees of weight n with distinct leaves.

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A318485 Number of p-trees of weight 2n + 1 in which all outdegrees are odd.

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