A318144 -id:A318144 - cp's OEIS frontend

A260845 a(n) = Sum_{k=0..n} (-1)^kP(n,k)k!, where P(n,k) is the number of partitions of n into k parts.

1, -1, 1, -5, 21, -105, 635, -4507, 36457, -330971, 3334377, -36913947, 445426739, -5818545721, 81805507069, -1231690773053, 19772941871385, -337146625794753, 6085005877228943, -115897323408009187, 2323090928155541677, -48883768421712917555, 1077440388662366900397

Offset: 0

Peter Luschny, Aug 01 2015

sign

Alois P. Heinz, Table of n, a(n) for n = 0..449

Cf. A008284, A101880.

Row sums of A318144.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>1,
      b(n, i-1), 0)+expand(b(n-i, min(n-i, i))*x))
    end:
a:= n-> (p-> add(i!*coeff(p, x, i)*(-1)^i, i=0..n))(b(n$2)):
seq(a(n), n=0..27);  # Alois P. Heinz, Sep 18 2019

CoefficientList[ Series[ Sum[ n!(-x)^n / Product[1 - x^k, {k, n}], {n, 0, 22}], {x, 0, 22}], x]

from sage.combinat.partition import number_of_partitions_length
[sum([(-1)^k*number_of_partitions_length(n,k)*factorial(k) for k in (0..n)]) for n in (0..22)]

G.f.: Sum(n!*(-x)^n/Product(1-x^k, k=1..n), n=1..infinity).

A327028 T(n, k) = k! * Sum_{d|n} phi(d) * A008284(n/d, k) for n >= 1, T(0, k) = 0^k. Triangle read by rows for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 2, 6, 0, 4, 6, 6, 24, 0, 5, 4, 12, 24, 120, 0, 6, 12, 24, 48, 120, 720, 0, 7, 6, 24, 72, 240, 720, 5040, 0, 8, 16, 36, 144, 360, 1440, 5040, 40320, 0, 9, 12, 54, 144, 600, 2160, 10080, 40320, 362880

Offset: 0

Peter Luschny, Aug 20 2019

			[0] 1
[1] 0, 1
[2] 0, 2,  2
[3] 0, 3,  2,  6
[4] 0, 4,  6,  6,  24
[5] 0, 5,  4, 12,  24, 120
[6] 0, 6, 12, 24,  48, 120,  720
[7] 0, 7,  6, 24,  72, 240,  720,  5040
[8] 0, 8, 16, 36, 144, 360, 1440,  5040, 40320
[9] 0, 9, 12, 54, 144, 600, 2160, 10080, 40320, 362880

Cf. A008284, A318144, A000142 (main diagonal), A327025 (row sums), A327029.

A327028 := (n,k) -> `if`(n=0, 1, k!*add(phi(d)*A008284(n/d, k), d = divisors(n))):
seq(seq(A327028(n, k), k=0..n), n=0..9);

A327028[0 , k_] := 1;
A327028[n_, k_] := DivisorSum[n, EulerPhi[#] A318144[n/#, k] (-1)^k &];
Table[A327028[n, k], {n, 0,  9}, {k, 0,  n}] // Flatten

# uses[DivisorTriangle from A327029]
from sage.combinat.partition import number_of_partitions_length
def A318144Abs(n, k): return number_of_partitions_length(n, k)*factorial(k)
DivisorTriangle(euler_phi, A318144Abs, 10)

cp's OEIS Frontend

A260845 a(n) = Sum_{k=0..n} (-1)^kP(n,k)k!, where P(n,k) is the number of partitions of n into k parts.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A327028 T(n, k) = k! * Sum_{d|n} phi(d) * A008284(n/d, k) for n >= 1, T(0, k) = 0^k. Triangle read by rows for 0 <= k <= n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

SageMath

cp's OEIS Frontend

A260845 a(n) = Sum_{k=0..n} (-1)^k*P(n,k)*k!, where P(n,k) is the number of partitions of n into k parts.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A327028 T(n, k) = k! * Sum_{d|n} phi(d) * A008284(n/d, k) for n >= 1, T(0, k) = 0^k. Triangle read by rows for 0 <= k <= n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

SageMath

A260845 a(n) = Sum_{k=0..n} (-1)^kP(n,k)k!, where P(n,k) is the number of partitions of n into k parts.