A066189 -id:A066189 - cp's OEIS frontend

A344088 Flattened tetrangle of reversed strict integer partitions sorted first by sum, then colexicographically.

1, 2, 1, 2, 3, 1, 3, 4, 2, 3, 1, 4, 5, 1, 2, 3, 2, 4, 1, 5, 6, 1, 2, 4, 3, 4, 2, 5, 1, 6, 7, 1, 3, 4, 1, 2, 5, 3, 5, 2, 6, 1, 7, 8, 2, 3, 4, 1, 3, 5, 4, 5, 1, 2, 6, 3, 6, 2, 7, 1, 8, 9, 1, 2, 3, 4, 2, 3, 5, 1, 4, 5, 1, 3, 6, 4, 6, 1, 2, 7, 3, 7, 2, 8, 1, 9, 10

Offset: 0

Gus Wiseman, May 12 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (12)(3)
  4: (13)(4)
  5: (23)(14)(5)
  6: (123)(24)(15)(6)
  7: (124)(34)(25)(16)(7)
  8: (134)(125)(35)(26)(17)(8)
  9: (234)(135)(45)(126)(36)(27)(18)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
The non-strict version is A080576.
Taking lex instead of colex gives A246688 (non-reversed: A344086).
The non-reversed version is A344087.
Taking revlex instead of colex gives A344089 (non-reversed: A118457).
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344091.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Table[Sort[Reverse/@Select[IntegerPartitions[n],UnsameQ@@#&],colex],{n,0,10}]

A116682 Sum of the odd parts in all partitions of n into distinct parts.

Original entry on oeis.org

0, 1, 0, 4, 4, 9, 10, 17, 26, 38, 50, 66, 92, 116, 154, 203, 264, 326, 416, 514, 644, 802, 986, 1198, 1474, 1784, 2156, 2608, 3124, 3728, 4454, 5286, 6266, 7420, 8736, 10279, 12062, 14106, 16472, 19214, 22330, 25914, 30032, 34714, 40058, 46208, 53136

Offset: 0

Emeric Deutsch, Feb 22 2006

nonn

			a(9)=38 because in the partitions of 9 into distinct parts, namely, [9],[8,1],[7,2],[6,3],[6,2,1],[5,4],[5,3,1] and [4,3,2], the sum of the odd parts is 9+1+7+3+1+5+5+3+1+3=38.

David A. Corneth, Table of n, a(n) for n = 0..10000

Cf. A000009, A000700, A066189, A116681, A116683, A116684.

f:=product(1+x^j,j=1..70)*sum((2*j-1)*x^(2*j-1)/(1+x^(2*j-1)),j=1..40): fser:=series(f,x=0,60): seq(coeff(fser,x,n),n=0..50);

d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]
Map[Total[Select[Flatten[d[#]], OddQ]] &, -1 + Range[30]]  (* Peter J. C. Moses, Mar 14 2014 *)
(* or *)
CoefficientList[Series[QPochhammer[-1, x]*(1 + EllipticTheta[2, 0, x]^4 - EllipticTheta[4, 0, x]^4)/48, {x, 0, 100}], x] (* Vaclav Kotesovec, Jun 24 2025 *)

a(n) = Sum_{k=0..n} k*A116681(n,k).
G.f.: (Product_{j>=1} 1+x^j)*(Sum_{j>=1} (2*j-1)*x^(2*j-1)/(1+x^(2*j-1))).
a(n) + A116684(n) = A066189(n) = n*A000009(n). - Vaclav Kotesovec, Jun 24 2025
a(n) = Sum_{k=0..floor(n/2)} A000700(n-2*k) * A000009(2*k) * (n - 2*k). - David A. Corneth, Jun 24 2025

A116684 Sum of the even parts in all partitions of n into distinct parts.

Original entry on oeis.org

0, 0, 2, 2, 4, 6, 14, 18, 22, 34, 50, 66, 88, 118, 154, 202, 248, 320, 412, 512, 636, 794, 972, 1194, 1454, 1766, 2134, 2576, 3092, 3696, 4426, 5254, 6214, 7364, 8672, 10196, 11986, 14014, 16360, 19084, 22190, 25746, 29860, 34516, 39846, 45952, 52848

Offset: 0

Emeric Deutsch, Feb 22 2006

nonn

			a(9)=34 because in the partitions of 9 into distinct parts, namely, [9],[8,1],[7,2],[6,3],[6,2,1],[5,4],[5,3,1] and [4,3,2], the sum of the even parts is 8+2+6+6+2+4+4+2=34.

David A. Corneth, Table of n, a(n) for n = 0..10000

Cf. A000009, A000700, A066189, A116681, A116682, A116683.

f:=2*product(1+x^j,j=1..60)*sum((j*x^(2*j)/(1+x^(2*j)),j=1..35)): fser:=series(f,x=0,55): seq(coeff(fser,x,n),n=0..50);

d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]
Map[Total[Select[Flatten[d[#]], EvenQ]] &, -1 + Range[30]]  (* Peter J. C. Moses, Mar 14 2014 *)
(* or *)
CoefficientList[Series[QPochhammer[-1, x]*(EllipticTheta[3, 0, x]^4 + EllipticTheta[4, 0, x]^4 - 2)/48, {x, 0, 100}], x] (* Vaclav Kotesovec, Jun 24 2025 *)

a(n) = Sum_{k>=0} k*A116683(n,k).
G.f.: 2*(Product_{j>=1} 1+x^j)*(Sum_{j>=1} j*x^(2*j)/(1+x^(2*j))).
A116682(n) + a(n) = A066189(n) = n*A000009(n). - Vaclav Kotesovec, Jun 24 2025
a(n) = Sum_{k=0..floor(n/2)} A000700(n-2*k) * A000009(2*k) * (2*k). - David A. Corneth, Jun 24 2025

A277029 Convolution of A000203 and A000009.

Original entry on oeis.org

0, 1, 4, 8, 16, 25, 42, 61, 90, 130, 178, 242, 332, 436, 566, 747, 952, 1210, 1540, 1926, 2400, 2994, 3674, 4506, 5526, 6708, 8108, 9808, 11768, 14080, 16850, 20022, 23738, 28128, 33152, 39015, 45854, 53662, 62696, 73166, 85118, 98826, 114636, 132586, 153102

Offset: 0

Vaclav Kotesovec, Sep 25 2016

nonn

Apart from initial zero this is the convolution of A340793 and A036469. - Omar E. Pol, Feb 16 2021

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A066186 (convolution of A000203 and A000041).
Cf. A276432 (convolution of A000203 and A000219).
Cf. A066189, A036469, A340793.

Table[Sum[DivisorSigma[1, k] * PartitionsQ[n-k], {k,1,n}], {n,0,50}]
nmax = 50; CoefficientList[Series[Sum[j*x^j/(1-x^j), {j, 1, nmax}]*Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{j>=1} (j*x^j/(1-x^j))*Product_{k>=1} (1+x^k).

a(n) ~ 2*n*A000009(n) ~ exp(Pi*sqrt(n/3)) * n^(1/4) / (2*3^(1/4)).

A306923 Sum over all partitions of n into distinct parts of the bitwise AND of the parts.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 7, 7, 12, 18, 18, 22, 23, 25, 19, 26, 37, 49, 55, 64, 67, 78, 80, 93, 101, 110, 106, 122, 114, 129, 136, 158, 197, 237, 256, 287, 311, 337, 367, 403, 424, 453, 492, 525, 571, 638, 684, 754, 809, 853, 896, 955, 995, 1075, 1149, 1226, 1295, 1412

Offset: 0

Alois P. Heinz, Mar 16 2019

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Bitwise operation
Wikipedia, Commutative property
Wikipedia, Identity element
Wikipedia, Partition (number theory)
Wikipedia, Truth table

Cf. A000009, A066189, A306901, A306924, A306925.

b:= proc(n, i, r) option remember; `if`(i*(i+1)/2 `if`(i b(n$2, 2^ilog2(2*n)-1):
seq(a(n), n=0..57);

A306924 Sum over all partitions of n into distinct parts of the bitwise OR of the parts.

Original entry on oeis.org

0, 1, 2, 6, 7, 13, 20, 35, 42, 58, 75, 104, 133, 176, 233, 327, 402, 500, 616, 762, 916, 1112, 1329, 1640, 1967, 2350, 2787, 3352, 3960, 4706, 5571, 6706, 7922, 9374, 10982, 12933, 15090, 17578, 20322, 23692, 27391, 31626, 36308, 41788, 47799, 54704, 62258

Offset: 0

Alois P. Heinz, Mar 16 2019

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Bitwise operation
Wikipedia, Commutative property
Wikipedia, Identity element
Wikipedia, Partition (number theory)
Wikipedia, Truth table

Cf. A000009, A066189, A306902, A306923, A306925.

b:= proc(n, i, r) option remember; `if`(i*(i+1)/2 `if`(i b(n$2, 0):
seq(a(n), n=0..49);

A306925 Sum over all partitions of n into distinct parts of the bitwise XOR of the parts.

Original entry on oeis.org

0, 1, 2, 6, 6, 11, 16, 35, 36, 46, 50, 84, 94, 130, 158, 285, 338, 424, 460, 616, 672, 810, 816, 1162, 1346, 1680, 1754, 2308, 2562, 3164, 3288, 4486, 5306, 6838, 7522, 9627, 11006, 13496, 14200, 17462, 19682, 24036, 25650, 30842, 33884, 40302, 41644, 48896

Offset: 0

Alois P. Heinz, Mar 16 2019

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Bitwise operation
Wikipedia, Commutative property
Wikipedia, Identity element
Wikipedia, Partition (number theory)
Wikipedia, Truth table

Cf. A000009, A066189, A067588, A067589, A306903, A306923, A306924.

b:= proc(n, i, r) option remember; `if`(i*(i+1)/2 `if`(i b(n$2, 0):
seq(a(n), n=0..51);

a(n) is odd <=> n in { A067589 }.

a(n) is odd <=> A067588(n) is odd.

A325515 Sum of sums of omegas of the parts over all strict integer partitions of n.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 8, 11, 14, 22, 29, 37, 50, 63, 81, 106, 129, 160, 203, 246, 303, 373, 449, 541, 654, 782, 932, 1116, 1322, 1559, 1848, 2167, 2537, 2978, 3470, 4041, 4706, 5449, 6303, 7291, 8402, 9665, 11117, 12744, 14592, 16708, 19062, 21730, 24757, 28141

Offset: 0

Gus Wiseman, May 07 2019

nonn

Also omega of the product of products of parts over all strict integer partitions of n.

The omega of n is A001222(n), the number of prime factors of n counted with multiplicity.

Cf. A001222, A003963, A015716, A015723, A022629, A056239, A066189, A112798, A147655, A246867, A325504, A325506.

Table[Sum[Total[PrimeOmega/@s],{s,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,30}]

a(n) = A001222(A325504(n)).

A304797 Expansion of x * (d/dx) Sum_{k>=0} k!x^(k(k+1)/2)/Product_{j=1..k} (1 - x^j).

Original entry on oeis.org

0, 1, 2, 9, 12, 25, 66, 91, 152, 243, 570, 715, 1212, 1729, 2702, 5265, 6960, 10489, 15318, 22363, 31100, 57771, 72534, 109411, 151032, 219025, 293930, 421281, 680820, 883369, 1256010, 1727971, 2396000, 3235419, 4447506, 5894875, 9266580, 11691001, 16380470, 21774753

Offset: 0

Ilya Gutkovskiy, May 18 2018

nonn

Sum of all parts of all compositions (ordered partitions) of n into distinct parts.

Index entries for sequences related to compositions

Cf. A001787, A032020, A066186, A066189, A097910, A303664.

b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
     `if`(k=0, `if`(n=0, 1, 0), b(n-k, k) +k*b(n-k, k-1)))
    end:
a:= n-> n*add(b(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..50);  # Alois P. Heinz, May 18 2018

nmax = 39; CoefficientList[Series[x D[Sum[k! x^(k (k + 1)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], x], {x, 0, nmax}], x]

a(n) = n*A032020(n).

A306919 Sum over all partitions of n into distinct parts of the power tower evaluation x^y^...^z, where x, y, ..., z are the parts in increasing order.

Original entry on oeis.org

1, 1, 2, 4, 5, 14, 24, 122, 318, 2417851639229258349414245, 14134776518227074636666380005943348126619871175004951664972849610340964762

Offset: 0

Alois P. Heinz, Mar 16 2019

nonn

			a(0) = 1 because the empty partition () has no parts, the exponentiation operator ^ is right-associative, and 1 is the right identity of exponentiation.
a(6) = 1^2^3 + 2^4 + 1^5 + 6 = 1 + 16 + 1 + 6 = 24.

Alois P. Heinz, Table of n, a(n) for n = 0..11
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Exponentiation
Wikipedia, Identity element
Wikipedia, Operator associativity
Wikipedia, Partition (number theory)

Cf. A022629, A066189, A306895, A306918.

d:= proc(l) local i; for i to nops(l)-1 do
       if l[i]=l[i+1] then return fi od; l
    end:
f:= l-> `if`(l=[], 1, l[1]^f(subsop(1=(), l))):
a:= n-> add(f(l), l=map(l->d(sort(l, `<`)), combinat[partition](n))):
seq(a(n), n=0..11);

d[l_] := Module[{i}, For[i = 1, i <= Length[l]-1 , i++, If[l[[i]] == l[[i+1]], Return[]]]; l];
f[l_] := If[l == {}, 1, l[[1]]^f[Delete[l, 1]]];
a[n_] := Sum[f[l], {l, Sort /@ Select[IntegerPartitions[n], Length@# == Length @ Union@#&]}];
a /@ Range[0, 11] (* Jean-François Alcover, May 03 2020, after Maple *)

cp's OEIS Frontend

A344088 Flattened tetrangle of reversed strict integer partitions sorted first by sum, then colexicographically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A116682 Sum of the odd parts in all partitions of n into distinct parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A116684 Sum of the even parts in all partitions of n into distinct parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A277029 Convolution of A000203 and A000009.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A306923 Sum over all partitions of n into distinct parts of the bitwise AND of the parts.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A306924 Sum over all partitions of n into distinct parts of the bitwise OR of the parts.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A306925 Sum over all partitions of n into distinct parts of the bitwise XOR of the parts.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A325515 Sum of sums of omegas of the parts over all strict integer partitions of n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A304797 Expansion of x * (d/dx) Sum_{k>=0} k!*x^(k*(k+1)/2)/Product_{j=1..k} (1 - x^j).

Views

Author

A304797 Expansion of x * (d/dx) Sum_{k>=0} k!x^(k(k+1)/2)/Product_{j=1..k} (1 - x^j).