A051749 -id:A051749 - cp's OEIS frontend

A001255 Squares of partition numbers.

1, 1, 4, 9, 25, 49, 121, 225, 484, 900, 1764, 3136, 5929, 10201, 18225, 30976, 53361, 88209, 148225, 240100, 393129, 627264, 1004004, 1575025, 2480625, 3833764, 5934096, 9060100, 13823524, 20839225, 31404816, 46812964, 69705801, 102880449, 151536100

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A000041, A000290, A304990, A200089.

Cf. A133042, A054440.

a001255 = (^ 2) . a000041  -- Reinhard Zumkeller, Nov 15 2015

seq(numbpart(k)^2, k=0..33); # Zerinvary Lajos, Jun 06 2007

Table[ PartitionsP[n]^2, {n, 0, 33}] (* Ray Chandler, Nov 14 2005 *)

a(n) = numbpart(n)^2; \\ Michel Marcus, May 01 2021

a(n) = A000041(n)^2.
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (48*n^2). - Vaclav Kotesovec, Dec 01 2015
Sum_{n>=1} 1/a(n) = A200089. - Amiram Eldar, May 01 2021
a(n) = A006907(n) + A051748(n) + A051749(n). - R. J. Mathar, Mar 09 2022
a(n) = [(x*y)^n] Product_{k>=1} 1 / ((1 - x^k) * (1 - y^k)). - Ilya Gutkovskiy, Apr 24 2025

Extended by Ray Chandler, Nov 14 2005

A006907 Number of zeros in character table of symmetric group S_n.

Original entry on oeis.org

0, 0, 1, 4, 10, 29, 55, 153, 307, 588, 1018, 2230, 3543, 6878, 11216, 20615, 33355, 57980, 90194, 155176, 239327, 395473, 604113, 970294, 1453749, 2323476, 3425849, 5349414, 7905133, 11963861, 17521274, 26472001, 38054619, 56756488, 81683457, 119005220, 170498286, 247619748

Offset: 1

Simon Plouffe and N. J. A. Sloane

J. McKay, personal communication to N. J. A. Sloane, circa 1991.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alexander R. Miller, On parity and characters of symmetric groups, preprint.
Alexander R. Miller, On parity and characters of symmetric groups, J. Combin. Theory Ser. A 162 (2019), 231-240.

Cf. A006908, A051748, A051749.

A006907 := n -> Sum(Irr(CharacterTable("Symmetric", n)), chi -> Number(chi, x->x=0)); # Eric M. Schmidt, Jul 13 2012, simplified Jul 26 2012

a[n_] := Count[FiniteGroupData[{"SymmetricGroup", n}, "CharacterTable"], 0, 2]; Array[a, 10] (* Jean-François Alcover, Oct 08 2016 *)

a(n) + A051748(n) + A051749(n) = A001255(n). - R. J. Mathar, Mar 09 2022

More terms from Vladeta Jovovic, May 20 2003
More terms from Eric M. Schmidt, Jul 13 2012
a(36)-a(38) found by Alexander R. Miller (see 2019 reference). - N. J. A. Sloane, Jul 07 2020

A051748 Number of character table entries of the symmetric group S_n which are < 0.

Original entry on oeis.org

0, 1, 2, 7, 13, 34, 72, 137, 249, 524, 953, 1679, 3106, 5270, 9398, 15666, 26284, 43409, 72861, 115940, 189476, 297929, 476904, 743094, 1174624, 1782368, 2787256, 4196505, 6413986, 9645485, 14553197, 21483398, 32243250, 47165359, 69606943

Offset: 1

JOHN MCKAY (mckay(AT)cs.concordia.ca), Dec 08 1999

Cf. A006907, A051749.

A051748 := n -> Sum(Irr(CharacterTable("Symmetric", n)), chi -> Number(chi, x->x<0)); # Eric M. Schmidt, Jul 14 2012, revised Sep 05 2012

a[n_] := Count[FiniteGroupData[{"SymmetricGroup", n}, "CharacterTable"], ?Negative, 2]; Array[a, 10] (* _Jean-François Alcover, Apr 03 2017 *)

More terms from Eric M. Schmidt, Jul 14 2012

cp's OEIS Frontend

A001255 Squares of partition numbers.

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A006907 Number of zeros in character table of symmetric group S_n.

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A051748 Number of character table entries of the symmetric group S_n which are < 0.

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