A078408 -id:A078408 - 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)).

A365828 Number of strict integer partitions of 2n not containing n.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 18, 27, 39, 55, 78, 108, 148, 201, 270, 359, 475, 623, 811, 1050, 1351, 1728, 2201, 2789, 3517, 4418, 5527, 6887, 8553, 10585, 13055, 16055, 19685, 24065, 29343, 35685, 43287, 52387, 63253, 76200, 91605, 109897, 131575, 157231, 187539

Offset: 0

Gus Wiseman, Sep 20 2023

nonn

			The a(0) = 1 through a(6) = 12 strict partitions:
  ()  (2)  (4)    (6)    (8)      (10)       (12)
           (3,1)  (4,2)  (5,3)    (6,4)      (7,5)
                  (5,1)  (6,2)    (7,3)      (8,4)
                         (7,1)    (8,2)      (9,3)
                         (5,2,1)  (9,1)      (10,2)
                                  (6,3,1)    (11,1)
                                  (7,2,1)    (5,4,3)
                                  (4,3,2,1)  (7,3,2)
                                             (7,4,1)
                                             (8,3,1)
                                             (9,2,1)
                                             (5,4,2,1)

The complement is counted by A111133.
For non-strict partitions we have A182616, complement A000041.
A000009 counts strict integer partitions.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A365827 counts strict partitions not of length 2, complement A140106.
Cf. A008967, A035363, A078408, A079122, A231429, A238628, A344415, A365659.

Table[Length[Select[IntegerPartitions[2n],UnsameQ@@#&&FreeQ[#,n]&]],{n,0,30}]

a(n) = A000009(2n) - A000009(n) + 1.

A282893 The difference between the number of partitions of 2n into odd parts (A000009) and the number of partitions of 2n into even parts (A035363).

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 4, 7, 10, 16, 22, 33, 45, 64, 87, 120, 159, 215, 283, 374, 486, 634, 814, 1049, 1335, 1700, 2146, 2708, 3390, 4243, 5276, 6552, 8095, 9989, 12266, 15044, 18375, 22409, 27235, 33049, 39974, 48281, 58148, 69923, 83871, 100452, 120027, 143214, 170515, 202731, 240567, 285073, 337195

Offset: 0

Robert G. Wilson v, Feb 24 2017

nonn

The even bisection of A282892. The other bisection is A078408.

			G.f. = x^3 + x^4 + 3*x^5 + 4*x^6 + 7*x^7 + 10*x^8 + 16*x^9 + 22*x^10 + 33*x^11 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A000041, A035294, A035363, A078408, A214264, A282892.

with(numtheory):
b:= proc(n, t) option remember; `if`(n=0, 1, add(add(`if`(
      (d+t)::odd, d, 0), d=divisors(j))*b(n-j, t), j=1..n)/n)
    end:
a:= n-> b(2*n, 0) -b(2*n, 1):
seq(a(n), n=0..80);  # Alois P. Heinz, Feb 24 2017

f[n_] := Length[ IntegerPartitions[n, All, 2Range[n] -1]] - Length[ IntegerPartitions[n, All, 2 Range[n]]]; Array[ f[2#] &, 52]
a[ n_] := SeriesCoefficient[ Sum[ Sign @ SquaresR[1, 16 k + 1] x^k, {k, n}] / QPochhammer[x], {x, 0, n}]; (* Michael Somos, Feb 24 2017 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=1, n, issquare(16*k + 1)*x^k, A) / eta(x + A), n))}; /* Michael Somos, Feb 24 2017 */

a(n) = A282892(2n).
Expansion of (f(x^3, x^5) - 1) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Feb 24 2017
a(n) = A035294(n) - A000041(n). - Michael Somos, Feb 24 2017

A300440 Number of odd strict trees of weight n (all outdegrees are odd).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 5, 7, 11, 18, 27, 45, 75, 125, 207, 353, 591, 1013, 1731, 2984, 5122, 8905, 15369, 26839, 46732, 81850, 142932, 251693, 441062, 778730, 1370591, 2425823, 4281620, 7601359, 13447298, 23919512, 42444497, 75632126, 134454505, 240100289

Offset: 1

Gus Wiseman, Mar 05 2018

nonn

An odd strict tree of weight n is either a single node of weight n, or a finite odd-length sequence of at least 3 odd strict trees with strictly decreasing weights summing to n.

			The a(10) = 7 odd strict trees: 10, (721), (631), (541), (532), ((421)21), ((321)31).

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000009, A000992, A032305, A063834, A078408, A089259, A196545, A273873, A279785, A289501, A298118, A300301, A300352, A300353, A300436, A300439.

g[n_]:=g[n]=1+Sum[Times@@g/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&&UnsameQ@@#&]}];
Array[g,20]

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

A358823 Number of odd-length twice-partitions of n into partitions with all odd parts.

Original entry on oeis.org

0, 1, 1, 3, 3, 7, 10, 20, 29, 58, 83, 150, 230, 399, 605, 1037, 1545, 2547, 3879, 6241, 9437, 15085, 22622, 35493, 53438, 82943, 124157, 191267, 284997, 434634, 647437, 979293, 1452182, 2185599, 3228435, 4826596, 7112683, 10575699, 15530404, 22990800, 33651222

Offset: 0

Gus Wiseman, Dec 03 2022

nonn

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

Also the number of odd-length twice-partitions of n into strict partitions.

			The a(1) = 1 through a(6) = 10 twice-partitions with all odd parts:
  (1)  (11)  (3)        (31)        (5)              (33)
             (111)      (1111)      (311)            (51)
             (1)(1)(1)  (11)(1)(1)  (11111)          (3111)
                                    (3)(1)(1)        (111111)
                                    (11)(11)(1)      (3)(11)(1)
                                    (111)(1)(1)      (31)(1)(1)
                                    (1)(1)(1)(1)(1)  (11)(11)(11)
                                                     (111)(11)(1)
                                                     (1111)(1)(1)
                                                     (11)(1)(1)(1)(1)
The a(1) = 1 through a(6) = 10 twice-partitions into strict partitions:
  (1)  (2)  (3)        (4)        (5)              (6)
            (21)       (31)       (32)             (42)
            (1)(1)(1)  (2)(1)(1)  (41)             (51)
                                  (2)(2)(1)        (321)
                                  (3)(1)(1)        (2)(2)(2)
                                  (21)(1)(1)       (3)(2)(1)
                                  (1)(1)(1)(1)(1)  (4)(1)(1)
                                                   (21)(2)(1)
                                                   (31)(1)(1)
                                                   (2)(1)(1)(1)(1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences enumerating triangles of integer partitions

This is the odd-length case of A270995.
Requiring odd sums also gives A279374 aerated.
This is the case of A358824 with all odd parts.
A000009 counts partitions into odd parts.
A027193 counts partitions of odd length.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
A358334 counts twice-partitions into odd-length partitions.
Cf. A000041, A001970, A072233, A271619, A279785, A356932.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Length[#]]&&OddQ[Times@@Flatten[#]]&]],{n,0,10}]

R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)) - 1)); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ Andrew Howroyd, Dec 31 2022

G.f.: ((1/Product_{k>=1} (1-A000009(k)*x^k)) - (1/Product_{k>=1} (1+A000009(k)*x^k)))/2. - Andrew Howroyd, Dec 31 2022

Terms a(26) and beyond from Andrew Howroyd, Dec 31 2022

A365827 Number of strict integer partitions of n whose length is not 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 16, 20, 25, 30, 38, 45, 55, 66, 79, 93, 111, 130, 153, 179, 209, 242, 282, 325, 375, 432, 496, 568, 651, 742, 846, 963, 1094, 1240, 1406, 1589, 1795, 2026, 2282, 2567, 2887, 3240, 3634, 4072, 4557, 5094, 5692, 6351

Offset: 0

Gus Wiseman, Sep 20 2023

nonn

Also the number of strict integer partitions of n with no pair of distinct parts summing to n.

			The a(5) = 1 through a(13) = 12 strict partitions (A..D = 10..13):
  (5)  (6)    (7)    (8)    (9)    (A)     (B)     (C)     (D)
       (321)  (421)  (431)  (432)  (532)   (542)   (543)   (643)
                     (521)  (531)  (541)   (632)   (642)   (652)
                            (621)  (631)   (641)   (651)   (742)
                                   (721)   (731)   (732)   (751)
                                   (4321)  (821)   (741)   (832)
                                           (5321)  (831)   (841)
                                                   (921)   (931)
                                                   (5421)  (A21)
                                                   (6321)  (5431)
                                                           (6421)
                                                           (7321)

The complement is counted by A140106 shifted left.
Heinz numbers are A005117 \ A006881 = A005117 /\ A100959.
The non-strict version is A058984, complement A004526.
The case not containing n/2 is A365826, non-strict A365825.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A182616 counts partitions of 2n that do not contain n, strict A365828.
Cf. A008967, A035363, A078408, A365659.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[#]!=2&]],{n,0,30}]

a(n) = A000009(n) - A004526(n-1) for n > 0.

A358834 Number of odd-length twice-partitions of n into odd-length partitions.

Original entry on oeis.org

0, 1, 1, 3, 3, 8, 11, 24, 35, 74, 109, 213, 336, 624, 986, 1812, 2832, 5002, 7996, 13783, 21936, 37528, 59313, 99598, 158356, 262547, 415590, 684372, 1079576, 1759984, 2779452, 4491596, 7069572, 11370357, 17841534, 28509802, 44668402, 70975399, 110907748

Offset: 0

Gus Wiseman, Dec 04 2022

nonn

A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(1) = 1 through a(6) = 11 twice-partitions:
  (1)  (2)  (3)        (4)        (5)              (6)
            (111)      (211)      (221)            (222)
            (1)(1)(1)  (2)(1)(1)  (311)            (321)
                                  (11111)          (411)
                                  (2)(2)(1)        (21111)
                                  (3)(1)(1)        (2)(2)(2)
                                  (111)(1)(1)      (3)(2)(1)
                                  (1)(1)(1)(1)(1)  (4)(1)(1)
                                                   (111)(2)(1)
                                                   (211)(1)(1)
                                                   (2)(1)(1)(1)(1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for set partitions is A003712.
If the parts are also odd we get A279374.
The version for multiset partitions of integer partitions is the odd-length case of A356932, ranked by A026424 /\ A356935.
This is the odd-length case of A358334.
This is the odd-lengths case of A358824.
For odd sums instead of lengths we have A358826.
The case of odd sums also is the bisection of A358827.
A000009 counts partitions into odd parts.
A027193 counts partitions of odd length.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
Cf. A000041, A001970, A270995, A271619, A279785, A358823, A358837.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Length[#]]&&OddQ[Times@@Length/@#]&]],{n,0,10}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=Vec(P(n,1)-P(n,-1))/2); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ Andrew Howroyd, Dec 30 2022

G.f.: ((1/Product_{k>=1} (1-A027193(k)*x^k)) - (1/Product_{k>=1} (1+A027193(k)*x^k)))/2. - Andrew Howroyd, Dec 30 2022

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2022

A078410 Number of ways to partition 4*n+3 into distinct positive integers.

Original entry on oeis.org

2, 5, 12, 27, 54, 104, 192, 340, 585, 982, 1610, 2590, 4097, 6378, 9792, 14848, 22250, 32992, 48446, 70488, 101698, 145578, 206848, 291874, 409174, 570078, 789640, 1087744, 1490528, 2032290, 2757826, 3725410, 5010688, 6711480, 8953856

Offset: 0

N. J. A. Sloane, Dec 27 2002

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Bisection of A078408. Cf. A035294, A000009, A078409.

PartitionsQ[4*Range[0,40]+3] (* Harvey P. Dale, Sep 23 2013 *)

a(n) = t(4*n+3, 0), t as defined in A079211.

More terms from Reinhard Zumkeller, Dec 28 2002

A300575 Coefficient of x^n in (1+x)(1-x^3)(1+x^5)(1-x^7)(1+x^9)...

Original entry on oeis.org

1, 1, 0, -1, -1, 1, 1, -1, -2, 0, 2, 0, -3, -1, 3, 2, -3, -3, 3, 4, -3, -6, 2, 7, -1, -8, 0, 10, 2, -11, -4, 12, 7, -13, -10, 13, 13, -13, -17, 13, 22, -11, -26, 9, 31, -6, -36, 2, 41, 3, -46, -9, 51, 17, -55, -26, 59, 36, -62, -48, 63, 61, -64, -75, 64, 92, -60, -109, 55, 127, -48, -147, 37, 167

Offset: 0

Gus Wiseman, Mar 09 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000009, A010815, A027193, A067659, A078408, A081362, A220418, A290261, A292043, A292137, A300574.

CoefficientList[Series[QPochhammer[-x,-x^2],{x,0,100}],x]

O.g.f.: Product_{n >= 0} (1 + (-1)^n x^(2n+1)).

a(n) = Sum (-1)^k where the sum is over all strict integer partitions of n into odd parts and k is the number of parts not congruent to 1 modulo 4.

cp's OEIS Frontend

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

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

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A365828 Number of strict integer partitions of 2n not containing n.

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A282893 The difference between the number of partitions of 2n into odd parts (A000009) and the number of partitions of 2n into even parts (A035363).

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A300440 Number of odd strict trees of weight n (all outdegrees are odd).

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A358823 Number of odd-length twice-partitions of n into partitions with all odd parts.

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A365827 Number of strict integer partitions of n whose length is not 2.

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