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.

Showing 1-3 of 3 results.

A321896 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in e(u) * Product_i u_i!, where H is Heinz number, e is elementary symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 2, -3, 1, 0, -1, 1, -6, 3, 8, -6, 1, 0, 0, 1, 0, 1, 0, -2, 1, 0, 0, 2, -3, 1, 24, -30, -20, 15, 20, -10, 1, 0, 0, 0, -1, 1, -120, 90, 144, 40, -15, -90, -120, 45, 40, -15, 1, 0, -6, 0, 3, 8, -6, 1, 0, 0, -2, 3, 2, -4, 1, 0, 0, 0, 0, 1, 720
Offset: 1

Views

Author

Gus Wiseman, Nov 20 2018

Keywords

Comments

Row n has length A000041(A056239(n)).
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			Triangle begins:
     1
     1
    -1    1
     0    1
     2   -3    1
     0   -1    1
    -6    3    8   -6    1
     0    0    1
     0    1    0   -2    1
     0    0    2   -3    1
    24  -30  -20   15   20  -10    1
     0    0    0   -1    1
  -120   90  144   40  -15  -90 -120   45   40  -15    1
     0   -6    0    3    8   -6    1
     0    0   -2    3    2   -4    1
     0    0    0    0    1
   720 -840 -504 -420  630  504  210  280 -105 -210 -420  105   70  -21    1
     0    0    0    1    0   -2    1
For example, row 15 gives: 12e(32) = -2p(32) + 3p(221) + 2p(311) - 4p(2111) + p(11111).

Links

Crossrefs

Row sums are A036987.
Cf. A005651, A008480, A056239, A124794, A124795, A135278, A319193, A319225, A319226, A321742-A321765, A321897.

A321898 Sum of coefficients of power sums symmetric functions in h(y) * Product_i y_i! where h is homogeneous symmetric functions and y is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 24, 1, 4, 6, 120, 2, 720, 24, 12, 1, 5040, 4, 40320, 6, 48, 120, 362880, 2, 36, 720, 8, 24, 3628800, 12, 39916800, 1, 240, 5040, 144, 4, 479001600, 40320, 1440, 6, 6227020800, 48, 87178291200, 120, 24, 362880, 1307674368000, 2, 576, 36, 10080
Offset: 1

Views

Author

Gus Wiseman, Nov 20 2018

Keywords

Comments

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

Examples

			The sum of coefficients of 12h(32) = 2p(32) + 3p(221) + 2p(311) + 4p(2111) + p(11111) is a(15) = 12.

Links

Crossrefs

Row sums of A321897.
Cf. A005651, A008480, A056239, A124794, A124795, A135278, A296150, A319193, A321742-A321765.
Cf. A000142, A000720.

Programs

  • Mathematica
    f[p_, e_] := (PrimePi[p]!)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 10 2023 *)

Formula

Totally multiplicative with a(p) = primepi(p)! = A000142(A000720(p)). - Amiram Eldar, Sep 10 2023

A321933 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in h(u) * Product_i u_i!, where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and h is homogeneous symmetric functions.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 3, 1, 0, 1, 1, 0, 0, 1, 6, 3, 8, 6, 1, 0, 1, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 24, 30, 20, 15, 20, 10, 1, 0, 6, 0, 3, 8, 6, 1, 0, 0, 2, 3, 2, 4, 1, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Nov 23 2018

Keywords

Comments

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

Examples

			Tetrangle begins (zeros not shown):
  (1):  1
.
  (2):   1  1
  (11):     1
.
  (3):    2  3  1
  (21):      1  1
  (111):        1
.
  (4):     6  3  8  6  1
  (22):       1     2  1
  (31):          2  3  1
  (211):            1  1
  (1111):              1
.
  (5):     24 30 20 15 20 10  1
  (41):        6     3  8  6  1
  (32):           2  3  2  4  1
  (221):             1     2  1
  (311):                2  3  1
  (2111):                  1  1
  (11111):                    1
For example, row 14 gives: 12h(32) = 2p(32) + 3p(221) + 2p(311) + 4p(2111) + p(11111).

Links

Crossrefs

This is a regrouping of the triangle A321897.
Cf. A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.
Showing 1-3 of 3 results.

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

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