A005380 -id:A005380 - cp's OEIS frontend

A382339 Triangle read by rows: T(n,k) is the number of partitions of a 2-colored set of n objects into exactly k parts with 0 <= k <= n.

1, 0, 2, 0, 3, 3, 0, 4, 6, 4, 0, 5, 14, 9, 5, 0, 6, 22, 24, 12, 6, 0, 7, 37, 49, 34, 15, 7, 0, 8, 52, 92, 76, 44, 18, 8, 0, 9, 76, 157, 162, 103, 54, 21, 9, 0, 10, 100, 260, 302, 232, 130, 64, 24, 10, 0, 11, 135, 400, 554, 468, 302, 157, 74, 27, 11

Offset: 0

Peter Dolland, Mar 22 2025

			Triangle begins:
 0 : [1]
 1 : [0,  2]
 2 : [0,  3,   3]
 3 : [0,  4,   6,   4]
 4 : [0,  5,  14,   9,   5]
 5 : [0,  6,  22,  24,  12,   6]
 6 : [0,  7,  37,  49,  34,  15,   7]
 7 : [0,  8,  52,  92,  76,  44,  18,   8]
 8 : [0,  9,  76, 157, 162, 103,  54,  21,  9]
 9 : [0, 10, 100, 260, 302, 232, 130,  64, 24, 10]
10 : [0, 11, 135, 400, 554, 468, 302, 157, 74, 27, 11]
...

Alois P. Heinz, Rows n = 0..200, flattened

Row sums are A005380.
The 1-color case is A008284.
Cf. A381891.

b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, (n+1)*x^n,
      add(b(n-i*j, min(n-i*j, i-1))*binomial(i+j, j)*x^j, j=0..n/i)))
    end:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Mar 22 2025

b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, (n + 1)*x^n, Sum[b[n - i*j, Min[n - i*j, i - 1]]*Binomial[i + j, j]*x^j, {j, 0, n/i}]]];
T[n_, k_] := Coefficient[b[n, n], x, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 17 2025, after Alois P. Heinz *)

from sympy import binomial
from sympy.utilities.iterables import partitions
def t_row( n):
    if n == 0 : return [1]
    t = list( [0] * n)
    for p in partitions( n):
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= binomial( k + p[k], p[k])
        if s > 0 :
            t[s - 1] += fact
    return [0] + t

T(n,1) = n + 1 for n >= 1.
T(n,n) = n + 1.
T(n,k) = A381891(n,k) - A381891(n,k-1) for k >= 1.

A353065 Euler transform of odd primes.

Original entry on oeis.org

1, 3, 11, 32, 92, 239, 608, 1465, 3450, 7858, 17525, 38165, 81653, 171497, 354785, 723084, 1454642, 2889854, 5676607, 11031046, 21224439, 40453596, 76428636, 143192339, 266172016, 491072611, 899583306, 1636775949, 2958900040, 5316004485, 9494514599

Offset: 0

Ilya Gutkovskiy, Apr 21 2022

nonn

Cf. A005380, A030009, A030012, A065091.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*ithprime(d+1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 21 2022

nmax = 30; CoefficientList[Series[Product[1/(1 - x^k)^Prime[k + 1], {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Sum[d Prime[d + 1], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 30}]

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

A060285 Number of partitions of n objects of 2 colors with parts size >1.

Original entry on oeis.org

1, 0, 3, 4, 11, 18, 42, 70, 144, 248, 466, 802, 1442, 2444, 4247, 7116, 12030, 19878, 32938, 53670, 87429, 140680, 225815, 359100, 569157, 895224, 1402941, 2184662, 3388915, 5228458, 8035921, 12291710, 18732318, 28425342, 42981877, 64740330

Offset: 0

Vladeta Jovovic, Mar 23 2001

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Vaclav Kotesovec)
N. J. A. Sloane, Transforms

Cf. (row sums of) A060244, A054225, A005380.

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(k+1),{k,2,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 04 2015 *)

Euler transform of sequence [0, 3, 4, 5, 6, ...].
G.f.: Product_{k=2..infinity} 1/(1-x^k)^(k+1).
From Vaclav Kotesovec, Mar 09 2015: (Start)
For n>=2, a(n) = A005380(n-2) - 2*A005380(n-1) + A005380(n).
a(n) ~ 2^(1/36) * Zeta(3)^(37/36) * exp(1/12 - Pi^4/(432*Zeta(3)) + Pi^2 * n^(1/3) / (3*2^(4/3)*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * 3^(1/2) * Pi * n^(55/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... .
a(n) ~ (2*Zeta(3))^(2/3) * A005380(n) / n^(2/3).
(End)

Edited by Christian G. Bower, Jan 08 2004

A217194 Number of unlabeled simple graphs with n nodes of 2 colors whose components are path graphs.

Original entry on oeis.org

1, 2, 6, 16, 42, 106, 267, 656, 1602, 3868, 9270, 22048, 52140, 122580, 286798, 667944, 1549259, 3579738, 8242638, 18917600, 43286909, 98768820, 224768425, 510235760, 1155553468, 2611251662, 5888421059, 13252176464, 29768501556, 66749440076, 149415504274

Offset: 0

Geoffrey Critzer, Sep 27 2012

nonn

Here, a path graph is a connected graph with no cycles such that each node has degree at most two.

			a(3) = 16 because we have:
w w w; w w b; w b b; b b b;
w w-w; w w-b; w b-b; b w-w; b w-b; b b-b;
w-w-w; w-w-b; w-b-w; b-w-b; b-b-w; b-b-b, where the 2 colors are black b and white w.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf.: A217093, A005380.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*(2^(d-1)+
      2^(floor((d+1)/2)-1)), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 27 2012

nn=30;p=Product[1/(1- x^i)^(2^(i-1)+2^(Floor[(i+1)/2]-1)),{i,1,nn}];CoefficientList[Series[p,{x,0,nn}],x]

G.f.: Product_{i>=1} 1/(1-x^i)^(2^(i-1)+2^(floor((i+1)/2)-1)).

EULER transform of A005418.

A089351 Number of planar partitions of n with trace 4.

Original entry on oeis.org

1, 2, 6, 14, 33, 64, 127, 228, 404, 672, 1100, 1724, 2661, 3974, 5849, 8402, 11911, 16556, 22751, 30772, 41198, 54436, 71283, 92316, 118609, 150950, 190753, 239090, 297783, 368236, 452782, 553240, 672532, 812980, 978211, 1171144, 1396235

Offset: 4

Wouter Meeussen and Vladeta Jovovic, Dec 26 2003

Also number of partitions of n objects of 2 colors into 4 parts, each part containing at least one black object.

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (Ch. XI, exercise 5 and Ch. XII, exercise 5).

Column 4 of A089353. Cf. A000219, A005380, A005993 (trace 2), A050531 (trace 3).

G.f.: q^4*(q^12+q^10+2*q^9+4*q^8+2*q^7+4*q^6+2*q^5+4*q^4+2*q^3+q^2+1) / ((-1+q^4)^2*(-1+q^3)^2*(-1+q^2)^2*(-1+q)^2).

Edited and extended by Christian G. Bower, Jan 08 2004

A217201 Number of simple unlabeled graphs with n nodes of 2 colors whose components are cycles.

Original entry on oeis.org

1, 0, 0, 4, 6, 8, 23, 42, 83, 166, 324, 622, 1236, 2366, 4595, 8900, 17225, 33212, 64376, 124360, 240819, 466284, 904149, 1753782, 3407225, 6623274, 12892131, 25116456, 48987833, 95633480, 186891367, 365549578, 715661254, 1402246154, 2749778317, 5396266284

Offset: 0

Geoffrey Critzer, Sep 27 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A217194, A217093, A005380.

with (numtheory):
b:= n-> `if`(n<3, 0, add(phi(d)*2^(n/d)/(2*n), d=divisors(n))+
    `if`(irem(n, 2)=1, 2^((n-1)/2), 2^(n/2-1)+2^(n/2-2))):
a:= proc(n) option remember; local d, j; `if`(n=0, 1,
      add(add(d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 27 2012

Needs["Combinatorica`"]
a=Expand[Table[nn=n;CycleIndex[DihedralGroup[nn],s]/.Table[s[i]->2,{i,1,nn}],{n,1,30}]];
nn=30;p=Product[1/(1- x^i)^a[[i]],{i,3,nn}];CoefficientList[Series[p,{x,0,nn}],x]
(* Second program: *)
b[n_] := If[n < 3, 0, Sum[EulerPhi[d]*2^(n/d)/(2*n), {d, Divisors[n]}] +  If[Mod[n, 2] == 1, 2^((n - 1)/2), 2^(n/2 - 1) + 2^(n/2 - 2)]];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n - j], {j, 1, n}]/n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 05 2017, after Alois P. Heinz *)

EULER transform of 0,0,4,6,8,13,30,... A000029.

A299019 Expansion of Product_{k>=1} (1 - x^k)^(k+1).

Original entry on oeis.org

1, -2, -2, 2, 3, 6, -1, -2, -10, -14, -7, -2, 11, 26, 43, 30, 28, -6, -40, -92, -128, -132, -115, -48, 54, 200, 339, 484, 499, 476, 274, -32, -501, -998, -1539, -1924, -2042, -1838, -1139, 12, 1664, 3540, 5588, 7258, 8392, 8230, 6812, 3480, -1472, -8150, -15737, -23670, -30478

Offset: 0

Ilya Gutkovskiy, Aug 11 2018

sign

Convolution of A010815 and A073592.

Convolution inverse of A005380.

Cf. A005380, A010815, A073592, A092346.

nmax = 52; CoefficientList[Series[Product[(1 - x^k)^(k + 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 52; CoefficientList[Series[Exp[-Sum[(DivisorSigma[1, k] + DivisorSigma[2, k]) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[d (d + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 52}]

G.f.: exp(-Sum_{k>=1} (sigma_1(k) + sigma_2(k))*x^k/k).

A307975 G.f. A(x) satisfies: A(x) = x * exp(Sum_{k>=1} (A(x^k) + sigma(k)*x^k)/k).

Original entry on oeis.org

0, 1, 2, 6, 17, 52, 161, 524, 1739, 5929, 20562, 72471, 258596, 932897, 3395922, 12459900, 46028216, 171056252, 639072199, 2398886256, 9042816457, 34217811625, 129926976921, 494892472911, 1890469032715, 7240573075556, 27799085344845, 106970043377619, 412474047216418

Offset: 0

Ilya Gutkovskiy, May 08 2019

nonn

			G.f.: A(x) = x + 2*x^2 + 6*x^3 + 17*x^4 + 52*x^5 + 161*x^6 + 524*x^7 + 1739*x^8 + 5929*x^9 + 20562*x^10 + ...

Cf. A000041, A000081, A000203, A005380, A307976.

terms = 28; A[] = 0; Do[A[x] = x Exp[Sum[(A[x^k] + DivisorSigma[1, k] x^k)/k, {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 - x^k)^(a[k] + 1), {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 0, 28}]

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} 1/(1 - x^n)^(a(n)+1).

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d*(a(d) + 1) ) * a(n-k+1).

cp's OEIS Frontend

A382339 Triangle read by rows: T(n,k) is the number of partitions of a 2-colored set of n objects into exactly k parts with 0 <= k <= n.

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A353065 Euler transform of odd primes.

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Maple

Mathematica

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A060285 Number of partitions of n objects of 2 colors with parts size >1.

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Mathematica

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Extensions

A217194 Number of unlabeled simple graphs with n nodes of 2 colors whose components are path graphs.

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Maple

Mathematica

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A089351 Number of planar partitions of n with trace 4.

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Extensions

A217201 Number of simple unlabeled graphs with n nodes of 2 colors whose components are cycles.

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Maple

Mathematica

Formula

A299019 Expansion of Product_{k>=1} (1 - x^k)^(k+1).

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Mathematica

Formula

A307975 G.f. A(x) satisfies: A(x) = x * exp(Sum_{k>=1} (A(x^k) + sigma(k)*x^k)/k).

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