A066189 -id:A066189 - cp's OEIS frontend

A325537 Irregular triangle whose rows are the sorted combined parts of all strict integer partitions of n.

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

Offset: 1

Gus Wiseman, May 08 2019

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)} with multiset union {1,1,2,2,3,4,5,6}, which is row n = 6.
Triangle begins:
  1
  2
  1 2 3
  1 3 4
  1 2 3 4 5
  1 1 2 2 3 4 5 6
  1 1 2 2 3 4 4 5 6 7
  1 1 1 2 2 3 3 4 5 5 6 7 8
  1 1 1 2 2 2 3 3 3 4 4 5 5 6 6 7 8 9

Row lengths are A015723.
Row sums are A066189.
Row products are A325504.
Run-lengths of row n are row n of A325513.
Cf. A000009, A006128, A022629, A181821, A302247, A325515.

Table[Sort[Join@@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,10}]

A347060 Total number of 1's in the binary expansion of parts in all partitions of n into distinct parts.

Original entry on oeis.org

0, 1, 1, 4, 4, 7, 11, 15, 20, 28, 39, 48, 64, 80, 104, 134, 167, 203, 257, 311, 381, 470, 566, 680, 820, 981, 1168, 1394, 1650, 1946, 2300, 2700, 3161, 3705, 4315, 5026, 5845, 6769, 7827, 9049, 10424, 11992, 13784, 15801, 18088, 20702, 23620, 26922, 30665

Offset: 0

Alois P. Heinz, Aug 14 2021

			a(5) = 7 counts the 1's in [101], [100, 1], [11, 10].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000009, A000120, A066189, A066624, A318756, A319140.

h:= proc(n) option remember; add(i, i=Bits[Split](n)) end:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(n>i*(i+1)/2, 0, b(n, i-1)+(p-> p+
       [0, p[1]*h(i)])(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..60);

A278406 G.f.: x^2 * f''(x), where f(x) = Product_{k>=1} 1 / (1 - x^k).

Original entry on oeis.org

0, 0, 4, 18, 60, 140, 330, 630, 1232, 2160, 3780, 6160, 10164, 15756, 24570, 36960, 55440, 80784, 117810, 167580, 238260, 332640, 462924, 635030, 869400, 1174800, 1583400, 2113020, 2810808, 3706780, 4875480, 6363060, 8282208, 10711008, 13811820, 17710770

Offset: 0

Vaclav Kotesovec, Nov 21 2016

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000041, A066186, A066189, A278407.

nmax=60; CoefficientList[Series[x^2*D[Product[1/(1-x^k), {k, 1, nmax}], {x, 2}], {x, 0, nmax}], x]
nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]*Range[0, nmax]*(Range[0, nmax]-1)

a(n) = n*(n-1)*A000041(n).

A278407 G.f.: x^2 * f''(x), where f(x) = Product_{k>=1} (1 + x^k).

Original entry on oeis.org

0, 0, 2, 12, 24, 60, 120, 210, 336, 576, 900, 1320, 1980, 2808, 4004, 5670, 7680, 10336, 14076, 18468, 24320, 31920, 41118, 52624, 67344, 85200, 107250, 134784, 167832, 207872, 257520, 316200, 386880, 473088, 574464, 696150, 841680, 1012320, 1214784, 1455324

Offset: 0

Vaclav Kotesovec, Nov 21 2016

nonn

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000009, A066189, A278406.

g:= mul(1+x^k,k=1..100):
S1:= series(g,x,101):
S2:= series(x^2*diff(S1,x$2),x,101):
seq(coeff(S2,x,j),j=0..100); # Robert Israel, Nov 22 2016

nmax=60; CoefficientList[Series[x^2*D[Product[1+x^k, {k, 1, nmax}], {x, 2}], {x, 0, nmax}], x]
nmax=60; CoefficientList[Series[Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x]*Range[0, nmax]*(Range[0, nmax]-1)

a(n) = n*(n-1)*A000009(n).

A306918 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 decreasing order.

Original entry on oeis.org

1, 1, 2, 5, 7, 18, 36, 118, 265, 263212, 2217881, 152599933940, 542101086242752217003726400434973829461152534, 63340828764059520458379290673240751904836319648345

Offset: 0

Alois P. Heinz, Mar 16 2019

nonn

a(14) = 620606987...270037949 has 183231 decimal digits.

			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) = 3^2^1 + 4^2 + 5^1 + 6 = 9 + 16 + 5 + 6 = 36.

Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Exponentiation
Wikipedia, Identity element
Wikipedia, Operator associativity
Wikipedia, Partition (number theory)

Cf. A022629, A066189, A306884, A306919.

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..13);

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, ReverseSort /@ Select[IntegerPartitions[n], Length@# == Length@ Union@# &]}];
a /@ Range[0, 13] (* Jean-François Alcover, May 04 2020, after Maple *)

A270105 a(n) = Sum_{k=0..n} k*A000009(k).

Original entry on oeis.org

0, 1, 3, 9, 17, 32, 56, 91, 139, 211, 311, 443, 623, 857, 1165, 1570, 2082, 2728, 3556, 4582, 5862, 7458, 9416, 11808, 14736, 18286, 22576, 27760, 33976, 41400, 50280, 60820, 73300, 88084, 105492, 125967, 150015, 178135, 210967, 249265, 293785, 345445, 405337

Offset: 0

Vaclav Kotesovec, Mar 12 2016

nonn

Cf. A000009, A000070, A014153, A036469, A095944, A182738.

Partial sums of A066189.

Table[Sum[PartitionsQ[k]*k, {k, 0, n}], {n, 0, 50}]

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

G.f.: x*f'(x)/(1 - x), where f(x) = Product_{k>=1} (1 + x^k). - Ilya Gutkovskiy, Apr 13 2017

A279412 Expansion of Sum_{k>=1} prime(k)x^prime(k)/(1 + x^prime(k)) Product_{k>=1} (1 + x^prime(k)).

Original entry on oeis.org

0, 2, 3, 0, 10, 0, 14, 8, 9, 20, 11, 24, 26, 28, 30, 48, 34, 72, 57, 80, 84, 88, 115, 120, 125, 156, 135, 168, 203, 180, 279, 224, 297, 306, 315, 396, 407, 418, 507, 480, 574, 630, 645, 748, 720, 828, 893, 960, 1029, 1150, 1122, 1300, 1378, 1458, 1650, 1624, 1824, 1856, 2065, 2220, 2379, 2480, 2646, 2816, 2925

Offset: 1

Ilya Gutkovskiy, Apr 11 2017

nonn

Sum of all parts of all partitions of n into distinct primes.

			a(12) = 24 because we have [7, 5], [7, 3, 2] and 2*12 = 24.

Eric Weisstein's World of Mathematics, Prime Partition
Index entries for related partition-counting sequences

Cf. A000586, A024938, A066189, A276560.

nmax = 65; Rest[CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 + x^Prime[k]), {k, 1, nmax}] Product[1 + x^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 65; Rest[CoefficientList[Series[x D[Product[1 + x^Prime[k], {k, 1, nmax}], x], {x, 0, nmax}], x]]

G.f.: Sum_{k>=1} prime(k)*x^prime(k)/(1 + x^prime(k)) * Product_{k>=1} (1 + x^prime(k)).
G.f.: x*f'(x), where f(x) = Product_{k>=1} (1 + x^prime(k)).
a(n) = n*A000586(n).

A304907 Expansion of x * (d/dx) 1/(1 - Sum_{k>=1} x^k/(1 + x^k)).

Original entry on oeis.org

0, 1, 2, 9, 16, 35, 84, 161, 312, 639, 1240, 2354, 4536, 8593, 16128, 30360, 56672, 105213, 195174, 360582, 664040, 1220730, 2238324, 4095035, 7479552, 13636750, 24821108, 45114813, 81887008, 148438211, 268763160, 486082263, 878200416, 1585098372, 2858378368, 5149986275

Offset: 0

Ilya Gutkovskiy, May 20 2018

nonn

Sum of all parts of all Carlitz compositions (compositions without adjacent equal parts) of n.

Index entries for sequences related to compositions

Cf. A001787, A003242, A048272, A066186, A066189.

nmax = 35; CoefficientList[Series[x D[1/(1 - Sum[x^k/(1 + x^k), {k, 1, nmax}]), x], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[-(-1)^d, {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[n a[n], {n, 0, 35}]

a(n) = n*A003242(n).

cp's OEIS Frontend

A325537 Irregular triangle whose rows are the sorted combined parts of all strict integer partitions of n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A347060 Total number of 1's in the binary expansion of parts in all partitions of n into distinct parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A278406 G.f.: x^2 * f''(x), where f(x) = Product_{k>=1} 1 / (1 - x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A278407 G.f.: x^2 * f''(x), where f(x) = Product_{k>=1} (1 + x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A306918 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 decreasing order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A270105 a(n) = Sum_{k=0..n} k*A000009(k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A279412 Expansion of Sum_{k>=1} prime(k)*x^prime(k)/(1 + x^prime(k)) * Product_{k>=1} (1 + x^prime(k)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A304907 Expansion of x * (d/dx) 1/(1 - Sum_{k>=1} x^k/(1 + x^k)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A279412 Expansion of Sum_{k>=1} prime(k)x^prime(k)/(1 + x^prime(k)) Product_{k>=1} (1 + x^prime(k)).