cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-30 of 35 results. Next

A330417 Coefficient of e(y) in Sum_{k > 0, i > 0} x_i^k = p(1) + p(2) + p(3) + ..., where e is the basis of elementary symmetric functions, p is the basis of power-sum symmetric functions, and y is the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, -2, 1, 3, -3, -4, 1, 2, 4, 5, -4, -6, -5, -5, 1, 7, 5, -8, 5, 6, 6, 9, -5, 3, -7, -2, -6, -10, -12, 11, 1, -7, 8, -7, 9, -12, -9, 8, 6, 13, 14, -14, 7, 7, 10, 15, -6, 4, 7, -9, -8, -16, -7, 8, -7, 10, -11, 17, -21, -18, 12, -8, 1, -9, -16, 19, 9, -11
Offset: 1

Views

Author

Gus Wiseman, Dec 14 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Up to sign, a(n) is the number of acyclic spanning subgraphs of an undirected n-cycle whose component sizes are the prime indices of n.

Crossrefs

The unsigned version (except with a(1) = 1) is A319225.
The transition from p to e by Heinz numbers is A321752.
The transition from p to h by Heinz numbers is A321754.
Different orderings with and without signs and first terms are A115131, A210258, A263916, A319226, A330415.
Cf. A000041, A000110, A000258, A000670, A005651, A008480, A048994, A056239, A124794, A318762, A319191.

Programs

  • Mathematica
    Table[If[n==1,0,With[{tot=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]},(-1)^(tot-PrimeOmega[n])*tot*(PrimeOmega[n]-1)!/(Times@@Factorial/@FactorInteger[n][[All,2]])]],{n,30}]

Formula

a(n) = (-1)^(A056239(n) - Omega(n)) * A056239(n) * (Omega(n) - 1)! / Product c_i! where c_i is the multiplicity of prime(i) in the prime factorization of n.

A321915 Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in m(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.

Original entry on oeis.org

1, 2, -1, -1, 1, 3, -3, 1, -3, 5, -2, 1, -2, 1, 4, -2, -4, 4, -1, -2, 3, 2, -4, 1, -4, 2, 7, -7, 2, 4, -4, -7, 10, -3, -1, 1, 2, -3, 1, 5, -5, -5, 5, 5, -5, 1, -5, 9, 5, -7, -9, 9, -2, -5, 5, 11, -11, -8, 10, -2, 5, -7, -11, 14, 10, -14, 3, 5, -9, -8, 10, 12
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also the coefficient of e(v) in f(u), where f is forgotten symmetric functions and e is elementary symmetric functions.

Examples

			Tetrangle begins:
  (1):  1
.
  (2):   2 -1
  (11): -1  1
.
  (3):    3 -3  1
  (21):  -3  5 -2
  (111):  1 -2  1
.
  (4):     4 -2 -4  4 -1
  (22):   -2  3  2 -4  1
  (31):   -4  2  7 -7  2
  (211):   4 -4 -7 10 -3
  (1111): -1  1  2 -3  1
.
  (5):      5 -5 -5  5  5 -5  1
  (41):    -5  9  5 -7 -9  9 -2
  (32):    -5  5 11 11 -8 10 -2
  (221):    5 -7 11 14 10 14  3
  (311):    5 -9 -8 10 12 13  3
  (2111):  -5  9 10 14 13 17 -4
  (11111):  1 -2 -2  3  3 -4  1
For example, row 14 gives: m(32) = -5h(5) + 11h(32) + 5h(41) - 11h(221) - 8h(311) + 10h(2111) - 2h(11111).

Links

Crossrefs

This is a regrouping of the triangle A321748. Row sums are A155972.
Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A319191, A319193, A321912-A321935.

A321916 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in h(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and h is homogeneous symmetric functions.

Original entry on oeis.org

1, -1, 1, 0, 1, 1, -2, 1, 0, -1, 1, 0, 0, 1, -1, 1, 2, -3, 1, 0, 1, 0, -2, 1, 0, 0, 1, -2, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, 1, -2, -2, 3, 3, -4, 1, 0, -1, 0, 1, 2, -3, 1, 0, 0, -1, 2, 1, -3, 1, 0, 0, 0, 1, 0, -2, 1, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, -1, 1
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also the coefficient of h(v) in e(u).

Examples

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):  -1  1
  (11):     1
.
  (3):    1 -2  1
  (21):     -1  1
  (111):        1
.
  (4):    -1  1  2 -3  1
  (22):       1    -2  1
  (31):          1 -2  1
  (211):           -1  1
  (1111):              1
.
  (5):      1 -2 -2  3  3 -4  1
  (41):       -1     1  2 -3  1
  (32):          -1  2  1 -3  1
  (221):             1    -2  1
  (311):                1 -2  1
  (2111):                 -1  1
  (11111):                    1
For example, row 14 gives: h(32) = -e(32) + 2e(221) + e(311) - 3e(2111) + e(11111).

Links

Crossrefs

This is a regrouping of the triangle A321749. Row sums are A134286.
Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

A321919 Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in p(u), where u and v are integer partitions of n, H is Heinz number, h is homogeneous symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 2, -1, 0, 1, 3, -3, 1, 0, 2, -1, 0, 0, 1, 4, -2, -4, 4, -1, 0, 4, 0, -4, 1, 0, 0, 3, -3, 1, 0, 0, 0, 2, -1, 0, 0, 0, 0, 1, 5, -5, -5, 5, 5, -5, 1, 0, 4, 0, -2, -4, 4, -1, 0, 0, 6, -6, -3, 5, -1, 0, 0, 0, 4, 0, -4, 1, 0, 0, 0, 0, 3, -3, 1, 0, 0, 0, 0, 0, 2
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):   2 -1
  (11):     1
.
  (3):    3 -3  1
  (21):      2 -1
  (111):        1
.
  (4):     4 -2 -4  4 -1
  (22):       4    -4  1
  (31):          3 -3  1
  (211):            2 -1
  (1111):              1
.
  (5):      5 -5 -5  5  5 -5  1
  (41):        4    -2 -4  4 -1
  (32):           6 -6 -3  5 -1
  (221):             4    -4  1
  (311):                3 -3  1
  (2111):                  2 -1
  (11111):                    1
For example, row 14 gives: p(32) = 6h(32) - 6h(221) - 3h(311) + 5h(2111) - h(11111).

Links

Crossrefs

This is a regrouping of the triangle A321754.
Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

A321920 Tetrangle where T(n,H(u),H(v)) is the coefficient of e(v) in s(u), where u and v are integer partitions of n, H is Heinz number, e is elementary symmetric functions, and s is Schur functions.

Original entry on oeis.org

1, -1, 1, 1, 0, 1, -2, 1, -1, 1, 0, 1, 0, 0, -1, 1, 2, -3, 1, 0, 1, -1, 0, 0, 1, -1, -1, 1, 0, -1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, -2, -2, 3, 3, -4, 1, -1, 1, 2, -2, -1, 1, 0, 0, 1, -1, 1, -1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, -1, 0, 1, 0, 0, -1, 1, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):  -1  1
  (11):  1
.
  (3):    1 -2  1
  (21):  -1  1
  (111):  1
.
  (4):    -1  1  2 -3  1
  (22):       1 -1
  (31):    1 -1 -1  1
  (211):  -1     1
  (1111):  1
.
  (5):      1 -2 -2  3  3 -4  1
  (41):    -1  1  2 -2 -1  1
  (32):        1 -1  1 -1
  (221):      -1  1
  (311):    1 -1 -1     1
  (2111):  -1  1
  (11111):  1
For example, row 14 gives: s(32) = -e(32) + e(41) + e(221) - e(311).

Links

Crossrefs

Row sums are A134286. This is a regrouping of the triangle A321755.
Cf. A005651, A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, A319191, A319193, A321912-A321935.

A321921 Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in e(u), where u and v are integer partitions of n, H is Heinz number, s is Schur functions, and e is elementary symmetric functions.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 2, 1, 1, 2, 3, 3, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 2, 1, 2, 1, 0, 0, 0, 1, 1, 2, 1, 0, 1, 2, 3, 3, 3, 1, 1, 4, 5, 5, 6, 4
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			Tetrangle begins (zeroes not shown):
  (1): 1
.
  (2):    1
  (11): 1 1
.
  (3):       1
  (21):    1 1
  (111): 1 2 1
.
  (4):            1
  (22):     1   1 1
  (31):         1 1
  (211):    1 1 2 1
  (1111): 1 2 3 3 1
.
  (5):                 1
  (41):              1 1
  (32):          1   1 1
  (221):       1 2 1 2 1
  (311):         1 1 2 1
  (2111):    1 2 3 3 3 1
  (11111): 1 4 5 5 6 4 1
For example, row 14 gives: e(32) = s(221) + s(2111) + s(11111).

Links

Crossrefs

This is a regrouping of the triangle A321756.
Cf. A005651, A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, A319191, A319193, A321912-A321935.

A321922 Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in s(u), where u and v are integer partitions of n, H is Heinz number, h is homogeneous symmetric functions, and s is Schur functions.

Original entry on oeis.org

1, 1, 0, -1, 1, 1, 0, 0, -1, 1, 0, 1, -2, 1, 1, 0, 0, 0, 0, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, -1, 1, 0, -1, 1, 2, -3, 1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 1, -1, 1, -1, 0, 0, 1, -1, -1, 0, 1, 0, 0, -1, 1, 2, -2, -1, 1, 0
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):   1
  (11): -1  1
.
  (3):    1
  (21):  -1  1
  (111):  1 -2  1
.
  (4):     1
  (22):       1 -1
  (31):   -1     1
  (211):   1 -1 -1  1
  (1111): -1  1  2 -3  1
.
  (5):      1
  (41):    -1  1
  (32):       -1  1
  (221):       1 -1  1 -1
  (311):    1 -1 -1     1
  (2111):  -1  1  2 -2 -1  1
  (11111):  1 -2 -2  3  3 -4  1
For example, row 14 gives: s(32) = h(32) - h(41).

Links

Crossrefs

Row sums are A155972. This is a regrouping of the triangle A321758.
Cf. A005651, A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, A319191, A319193, A321912-A321935.

A321923 Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in h(u), where u and v are integer partitions of n, H is Heinz number, s is Schur functions, and h is homogeneous symmetric functions.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 2, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 2, 1, 0, 1, 2, 3, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 2, 2, 1, 1, 0, 0, 1, 2, 1, 0, 1, 0, 0, 1, 3, 3, 2, 3, 1, 0, 1, 4, 5, 5, 6, 4
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			Tetrangle begins (zeroes not shown):
  (1): 1
.
  (2):  1
  (11): 1 1
.
  (3):   1
  (21):  1 1
  (111): 1 2 1
.
  (4):    1
  (22):   1 1 1
  (31):   1   1
  (211):  1 1 2 1
  (1111): 1 2 3 3 1
.
  (5):     1
  (41):    1 1
  (32):    1 1 1
  (221):   1 2 2 1 1
  (311):   1 2 1   1
  (2111):  1 3 3 2 3 1
  (11111): 1 4 5 5 6 4 1
For example, row 14 gives: h(32) = s(5) + s(32) + s(41).

Links

Crossrefs

This is a regrouping of the triangle A321759.
Cf. A005651, A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, A319191, A319193, A321912-A321935.

A321926 Tetrangle where T(n,H(u),H(v)) is the coefficient of s(v) in p(u), where u and v are integer partitions of n, H is Heinz number, s is Schur functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 1, -1, 1, 1, 1, -1, 1, 1, 0, -1, 1, 2, 1, 1, 0, -1, 1, -1, 1, 2, -1, -1, 1, 1, -1, 0, 0, 1, 1, 0, 1, -1, -1, 1, 2, 3, 3, 1, 1, -1, 0, 0, 1, -1, 1, 1, 0, -1, 1, 0, 0, -1, 1, -1, 1, -1, 0, 1, -1, 1, 0, 1, 1, -2, 0, 1, 1, 1, -1, -1, 0, 1, 1, 1, 2, 1, -1, 0, -2
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):   1 -1
  (11):  1  1
.
  (3):    1 -1  1
  (21):   1    -1
  (111):  1  2  1
.
  (4):     1    -1  1 -1
  (22):    1  2 -1 -1  1
  (31):    1 -1        1
  (211):   1     1 -1 -1
  (1111):  1  2  3  3  1
.
  (5):      1 -1        1 -1  1
  (41):     1    -1  1       -1
  (32):     1 -1  1 -1     1 -1
  (221):    1     1  1 -2     1
  (311):    1  1 -1 -1     1  1
  (2111):   1  2  1 -1    -2 -1
  (11111):  1  4  5  5  6  4  1
For example, row 14 gives: p(32) = s(5) + s(32) - s(41) - s(221) + s(2111) - s(11111).

Links

Crossrefs

Row sums are A317552. This is a regrouping of the triangle A321765.
Cf. A005651, A008480, A056239, A124794, A124795, A153452, A215366, A296188, A300121, A319191, A319193, A321912-A321935.

A321927 Tetrangle where T(n,H(u),H(v)) is the coefficient of m(v) in f(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and f is forgotten symmetric functions.

Original entry on oeis.org

1, -1, 0, 1, 1, 1, 0, 0, -2, -1, 0, 1, 1, 1, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0, 1, 0, 0, -3, -2, -2, -1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -2, -1, 0, 0, 0, 0, 0, -2, 0, -1, 0, 0, 0, 0, 3, 1, 2, 1, 0, 0, 0, 3, 2, 1, 0, 1, 0, 0, -4, -3, -3, -2, -2, -1, 0
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also the coefficient of f(v) in m(u).

Examples

			Tetrangle begins (zeroes not shown):
  (1):  1
.
  (2):  -1
  (11):  1  1
.
  (3):    1
  (21):  -2 -1
  (111):  1  1  1
.
  (4):    -1
  (22):    1  1
  (31):    2     1
  (211):  -3 -2 -2 -1
  (1111):  1  1  1  1  1
.
  (5):      1
  (41):    -2 -1
  (32):    -2    -1
  (221):    3  1  2  1
  (311):    3  2  1     1
  (2111):  -4 -3 -3 -2 -2 -1
  (11111):  1  1  1  1  1  1  1
For example, row 14 gives: f(32) = -2m(5) - m(32).

Links

Crossrefs

This is a regrouping of the triangle A321886.
Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.
Previous Showing 21-30 of 35 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.