A153452 -id:A153452 - cp's OEIS frontend

A093641 Numbers of form 2^i * prime(j), i>=0, j>0, together with 1.

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 19, 20, 22, 23, 24, 26, 28, 29, 31, 32, 34, 37, 38, 40, 41, 43, 44, 46, 47, 48, 52, 53, 56, 58, 59, 61, 62, 64, 67, 68, 71, 73, 74, 76, 79, 80, 82, 83, 86, 88, 89, 92, 94, 96, 97, 101, 103, 104, 106, 107, 109, 112

Offset: 1

Reinhard Zumkeller, Apr 07 2004

a(n) is either 1, prime, or of form 2a(m), m

1 and Heinz numbers of hook integer partitions. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). A hook is a partition of the form (n,1,1,...,1). - Gus Wiseman, Sep 15 2018

Numbers whose odd part is noncomposite. - Peter Munn, Aug 06 2020

			55 is not a member, as 5*11 is not of the form 2^i * prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

A093640(a(n)) = A000005(a(n)); A000040 and A000079 are subsequences.
A105440 is a subsequence, see also A105442. - Reinhard Zumkeller, Apr 09 2005
Cf. A078822, A007088.
Complement of A105441; A001221(a(n))<=2; A005087(a(n))<=1; A087436(a(n))<=1.
See also A105442.
Union of A038550 and A000079, see also A008578.
Cf. A082733, A153452, A296188, A296561, A300121, A304438, A305940, A317554.
Cf. A000265 (odd part), A008578 (noncomposite).

a093641 n = a093641_list !! (n-1)
a093641_list = filter ((<= 2) . a001227) [1..]
-- Reinhard Zumkeller, May 01 2012

hookQ[n_]:=MatchQ[DeleteCases[FactorInteger[n],{2,}],{}|{{,1}}];
Select[Range[100],hookQ] (* Gus Wiseman, Sep 15 2018 *)

upTo(lim)=my(v=List([1])); for(e=0, log(lim)\log(2), forprime(p=2, lim>>e, listput(v,p<Charles R Greathouse IV, Aug 21 2011

isok(m) = my(k=m/2^valuation(m,2)); (k == 1) || isprime(k); \\ Michel Marcus, Mar 16 2023

from sympy import primepi
def A093641(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n-1+x-sum(primepi(x>>i) for i in range(x.bit_length()))
    return bisection(f,n,n) # Chai Wah Wu, Feb 02 2025

A001227(a(n)) <= 2. - Reinhard Zumkeller, May 01 2012

Number A(x) of a(n) not exceeding x equals 1 + pi(x) + pi(x/2) + pi(x/4) + ..., where pi(x) is the number of primes <= x. If x goes to infinity, A(x)~2*x/log(x) and a(n)~n*log(n)/2 (n-->infinity). - Vladimir Shevelev, Feb 06 2014

A053529 a(n) = n! * number of partitions of n.

Original entry on oeis.org

1, 1, 4, 18, 120, 840, 7920, 75600, 887040, 10886400, 152409600, 2235340800, 36883123200, 628929100800, 11769069312000, 230150688768000, 4833164464128000, 105639166144512000, 2464913876705280000, 59606099200327680000, 1525429559126753280000, 40464026199993876480000

Offset: 0

N. J. A. Sloane, Jan 16 2000

Commuting permutations: number of ordered pairs (g, h) in Sym(n) such that gh = hg.
Equivalently sum of the order of all normalizers of all cyclic subgroups of Sym(n). - Olivier Gérard, Apr 04 2012
From Gus Wiseman, Jan 16 2019: (Start)
Also the number of Young tableaux with distinct entries from 1 to n, where a Young tableau is an array obtained by replacing the dots in the Ferrers diagram of an integer partition of n with positive integers. For example, the a(3) = 18 tableaux are:
  123  213  132  312  231  321
.
  12   21   13   31   23   32
  3    3    2    2    1    1
.
  1  2  1  3  2  3
  2  1  3  1  3  2
  3  3  2  2  1  1
(End)

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.12, solution.

T. D. Noe, Table of n, a(n) for n = 0..200
M. Holloway, M. Shattuck, Commuting pairs of functions on a finite set, PU.M.A, Volume 24 (2013), Issue No. 1.
M. Holloway, M. Shattuck, Commuting pairs of functions on a finite set, Research Gate, 2015.
R. P. Stanley, Pairs with equal squares, Problem 10654, Amer. Math. Monthly, 107 (April 2000), solution p. 368.
Wikipedia, Young tableau

Column k=2 of A362827.
Cf. A000041, A072169, A061256.
Sequences counting pairs of functions from an n-set to itself: A053529, A181162, A239749-A239785, A239836-A239841.
Cf. A000085, A117433, A153452, A296188, A323295, A323434, A323436.

a:= func< n | NumberOfPartitions(n)*Factorial(n) >; [ a(n) : n in [0..25]]; // Vincenzo Librandi, Jan 17 2019

seq(count(Permutation(n))*count(Partition(n)),n=1..20); # Zerinvary Lajos, Oct 16 2006
with(combinat): A053529 := proc(n): n! * numbpart(n) end: seq(A053529(n), n=0..20); # Johannes W. Meijer, Jul 28 2016

Table[PartitionsP[n] n!, {n, 0, 20}] (* T. D. Noe, Jun 19 2012 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, x^k/(1-x^k)/k)))) \\ Joerg Arndt, Apr 16 2010

N=66; x='x+O('x^N); Vec(serlaplace(sum(n=0, N, x^n/prod(k=1,n,1-x^k)))) \\ Joerg Arndt, Jan 29 2011

a(n) = n!*numbpart(n); \\ Michel Marcus, Jul 28 2016

from math import factorial
from sympy import npartitions
def A053529(n): return factorial(n)*npartitions(n) # Chai Wah Wu, Jul 10 2023

E.g.f: Sum_{n>=0} x^n/(Product_{k=1..n} 1-x^k) = exp(Sum_{n>=1} (x^n/n)/(1-x^n)). - Joerg Arndt, Jan 29 2011
a(n) = Sum{k=1..n} (((n-1)!/(n-k)!)*sigma(k)*a(n-k)), n > 0, and a(0)=1. See A274760. - Johannes W. Meijer, Jul 28 2016
a(n) ~ sqrt(Pi/6)*exp(sqrt(2/3)*Pi*sqrt(n))*n^n/(2*exp(n)*sqrt(n)). - Ilya Gutkovskiy, Jul 28 2016

A000898 a(n) = 2(a(n-1) + (n-1)a(n-2)) for n >= 2 with a(0) = 1.

Original entry on oeis.org

1, 2, 6, 20, 76, 312, 1384, 6512, 32400, 168992, 921184, 5222208, 30710464, 186753920, 1171979904, 7573069568, 50305536256, 342949298688, 2396286830080, 17138748412928, 125336396368896, 936222729254912, 7136574106003456, 55466948299223040, 439216305474605056, 3540846129311916032

Offset: 0

N. J. A. Sloane

Number of solutions to the rook problem on a 2n X 2n board having a certain symmetry group (see Robinson for details).
Also the value of the n-th derivative of exp(x^2) evaluated at 1. - N. Calkin, Apr 22 2010
For n >= 1, a(n) is also the sum of the degrees of the irreducible representations of the group of n X n signed permutation matrices (described in sequence A066051). The similar sum for the "ordinary" symmetric group S_n is in sequence A000085. - Sharon Sela (sharonsela(AT)hotmail.com), Jan 12 2002
It appears that this is also the number of permutations of 1, 2, ..., n+1 such that each term (after the first) is within 2 of some preceding term. Verified for n+1 <= 6. E.g., a(4) = 20 because of the 24 permutations of 1, 2, 3, 4, the only ones not permitted are 1, 4, 2, 3; 1, 4, 3, 2; 4, 1, 2, 3; and 4, 1, 3, 2. - Gerry Myerson, Aug 06 2003
Hankel transform is A108400. - Paul Barry, Feb 11 2008
From Emeric Deutsch, Jun 19 2010: (Start)
Number of symmetric involutions of [2n]. Example: a(2)=6 because we have 1234, 2143, 1324, 3412, 4231, and 4321. See the Egge reference, pp. 419-420.
Number of symmetric involutions of [2n+1]. Example: a(2)=6 because we have 12345, 14325, 21354, 45312, 52341, and 54321. See the Egge reference, pp. 419-420.
(End)
Binomial convolution of sequence A000085: a(n) = Sum_{k=0..n} binomial(n,k)*A000085(k)*A000085(n-k). - Emanuele Munarini, Mar 02 2016
The sequence can be obtained from the infinite product of 2 X 2 matrices [(1,N); (1,1)] by extracting the upper left terms, where N = (1, 3, 5, ...), the odd integers. - Gary W. Adamson, Jul 28 2016
Apparently a(n) is the number of standard domino tableaux of size 2n, where a domino tableau is a generalized Young tableau in which all rows and columns are weakly increasing and all regions are dominos. - Gus Wiseman, Feb 25 2018

			G.f. = 1 + 2*x + 6*x^2 + 20*x^3 + 76*x^4 + 312*x^5 + 1384*x^6 + 6512*x^7 + ...
The a(3) = 20 domino tableaux:
1 1 2 2 3 3
.
1 2 2 3 3
1
.
1 2 3 3   1 1 3 3   1 1 2 2
1 2       2 2       3 3
.
1 1 3 3   1 1 2 2
2         3
2         3
.
1 2 3   1 2 2   1 1 3
1 2 3   1 3 3   2 2 3
.
1 3 3   1 2 2
1       1
2       3
2       3
.
1 2   1 1   1 1
1 2   2 3   2 2
3 3   2 3   3 3
.
1 3   1 2   1 1
1 3   1 2   2 2
2     3     3
2     3     3
.
1 1
2
2
3
3
.
1
1
2
2
3
3 - _Gus Wiseman_, Feb 25 2018

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 5.1.4 Exer. 31.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..200
Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 10.
Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv:2406.06036 [math.RT], 2024. See p. 10.
C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating functions for generating trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
R. A. Brualdi, Shi-Mie Ma, Enumeration of involutions by descents and symmetric matrices, Eur. J. Combin. 43 (2015) 220-228
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
C.-O. Chow, Counting involutory, unimodal, and alternating signed permutations, Discr. Math. 306 (2006), 2222-2228.
R. Donaghey, Binomial self-inverse sequences and tangent coefficients, J. Combin. Theory, Series A, 21 (1976), 155-163.
Eric S. Egge, Restricted symmetric permutations, Ann. Combin., 11 (2007), 405-434.
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012
T. Halverson and M. Reeks, Gelfand Models for Diagram Algebras, arXiv:1302.6150 [math.RT], 2013-2014.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
Yen-chi R. Lin, Asymptotic Formula for Symmetric Involutions, arXiv:1310.0988 [math.CO], 2013.
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
J. Riordan, Letter to N. J. A. Sloane, Feb 03 1975 (with notes by njas)
D. P. Roberts and A. Venkatesh, Hurwitz monodromy and full number fields, 2014.
Index entries for sequences related to Hermite polynomials

Column k=2 of A294042 and A376826.
Row sums of A122832 and A185296.
Cf. A000085, A000712, A004003, A066051, A099390, A100861, A135401, A138178, A153452, A297388, A299699, A299926, A300056, A300060.

a000898 n = a000898_list !! n
a000898_list = 1 : 2 : (map (* 2) $
   zipWith (+) (tail a000898_list) (zipWith (*) [1..] a000898_list))
-- Reinhard Zumkeller, Oct 10 2011

# For Maple program see A000903.
seq(simplify((-I)^n*HermiteH(n, I)), n=0..25); # Peter Luschny, Oct 23 2015

a[n_] := Sum[ 2^k*StirlingS1[n, k]*BellB[k], {k, 0, n}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 17 2011, after Vladeta Jovovic *)
RecurrenceTable[{a[0]==1,a[1]==2,a[n]==2(a[n-1]+(n-1)a[n-2])},a,{n,30}] (* Harvey P. Dale, Aug 04 2012 *)
Table[Abs[HermiteH[n, I]], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 22 2015 *)
a[ n_] := Sum[ 2^(n - 2 k) n! / (k! (n - 2 k)!), {k, 0, n/2}]; (* Michael Somos, Oct 23 2015 *)

makelist((%i)^n*hermite(n,-%i),n,0,12); /* Emanuele Munarini, Mar 02 2016 */

{a(n) = if( n<0, 0, n! * polcoeff( exp(2*x + x^2 + x * O(x^n)), n))}; /* Michael Somos, Feb 08 2004 */

{a(n) = if( n<2, max(0, n+1), 2*a(n-1) + (2*n - 2) * a(n-2))}; /* Michael Somos, Feb 08 2004 */

my(x='x+O('x^66)); Vec(serlaplace(exp(2*x+x^2))) \\ Joerg Arndt, Oct 04 2013

{a(n) = sum(k=0, n\2, 2^(n - 2*k) * n! / (k! * (n - 2*k)!))}; /* Michael Somos, Oct 23 2015 */

a(n) = Sum_{m=0..n} |A060821(n,m)| = H(n,-i)*i^n, with the Hermite polynomials H(n,x); i.e., these are row sums of the unsigned triangle A060821.
E.g.f.: exp(x*(x + 2)).
a(n) = 2 * A000902(n) for n >= 1.
a(n) = Sum_{k=0..n} binomial(n,2k)*binomial(2k,k)*k!*2^(n-2k). - N. Calkin, Apr 22 2010
Binomial transform of A047974. - Paul Barry, May 09 2003
a(n) = Sum_{k=0..n} Stirling1(n, k)*2^k*Bell(k). - Vladeta Jovovic, Oct 01 2003
From Paul Barry, Aug 29 2005: (Start)
a(n) = Sum_{k=0..floor(n/2)} A001498(n-k, k) * 2^(n-k).
a(n) = Sum_{k=0..n} A001498((n+k)/2, (n-k)/2) * 2^((n+k)/2) * (1+(-1)^(n-k))/2. (End)
For asymptotics, see the Robinson paper. [This is disputed by Yen-chi R. Lin. See below, Sep 30 2013.]
a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * C(n,2*k) * (2*k)!/k!. - Paul Barry, Feb 11 2008
G.f.: 1/(1 - 2*x - 2*x^2/(1 - 2*x - 4*x^2/(1 - 2*x - 6*x^2/(1 - 2*x - 8*x^2/(1 - ... (continued fraction). - Paul Barry, Feb 25 2010
E.g.f.: exp(x^2 + 2*x) = Q(0); Q(k) = 1 + (x^2 + 2*x)/(2*k + 1 - (x^2 + 2*x)*(2*k + 1)/((x^2 + 2*x) + (2*k + 2)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
G.f.: 1/Q(0), where Q(k) = 1 + 2*x*k - x - x/(1 - 2*x*(k + 1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013
a(n) = (2*n/e)^(n/2) * exp(sqrt(2*n)) / sqrt(2*e) * (1 + sqrt(2/n)/3 + O(n^(-1))). - Yen-chi R. Lin, Sep 30 2013
0 = a(n)*(2*a(n+1) + 2*a(n+2) - a(n+3)) + a(n+1)*(-2*a(n+1) + a(n+2)) for all n >= 0. - Michael Somos, Oct 23 2015
a(n) = Sum_{k=0..floor(n/2)} 2^(n-k)*B(n, k), where B are the Bessel numbers A100861. - Peter Luschny, Jun 04 2021

More terms from Larry Reeves (larryr(AT)acm.org), Feb 21 2001
Initial condition a(0)=1 added to definition by Jon E. Schoenfield, Oct 01 2013
More terms from Joerg Arndt, Oct 04 2013

A296188 Number of normal semistandard Young tableaux whose shape is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 8, 1, 6, 12, 16, 6, 32, 32, 28, 1, 64, 16, 128, 24, 96, 80, 256, 8, 44, 192, 22, 80, 512, 96, 1024, 1, 288, 448, 224, 30, 2048, 1024, 800, 40, 4096, 400, 8192, 240, 168, 2304, 16384, 10, 360, 204, 2112, 672, 32768, 68, 832, 160, 5376, 5120

Offset: 1

Gus Wiseman, Feb 14 2018

nonn

A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(9) = 6 tableaux:
1 3   1 2   1 2   1 2   1 1   1 1
2 4   3 4   3 3   2 3   2 3   2 2

Richard P. Stanley, Enumerative Combinatorics Volume 2, Cambridge University Press, 1999, Chapter 7.10.

FindStat - Combinatorial Statistic Finder, Semistandard Young tableaux

Cf. A000085, A001222, A056239, A063834, A112798, A122111, A138178, A153452, A191714, A210391, A228125, A296150, A296560, A296561, A299202, A299966, A300056, A300121.

conj[y_List]:=If[Length[y]===0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
conj[n_Integer]:=Times@@Prime/@conj[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]];
ssyt[n_]:=If[n===1,1,Sum[ssyt[n/q*Times@@Cases[FactorInteger[q],{p_,k_}:>If[p===2,1,NextPrime[p,-1]^k]]],{q,Rest[Divisors[n]]}]];
Table[ssyt[conj[n]],{n,50}]

Let b(n) = Sum_{d|n, d>1} b(n * d' / d) where if d = Product_i prime(s_i)^m(i) then d' = Product_i prime(s_i - 1)^m(i) and prime(0) = 1. Then a(n) = b(conj(n)) where conj = A122111.

A321765 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of s(v) in p(u), where H is Heinz number, p is power sum symmetric functions, and s is Schur functions.

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 0, -1, 1, 0, -1, 1, -1, 1, 2, 1, 1, 2, -1, -1, 1, 1, -1, 0, 0, 1, 1, -1, 0, 0, 1, -1, 1, 1, 0, 1, -1, -1, 1, 0, -1, 0, 0, 1, 0, 0, -1, 1, -1, 1, 0, -1, 1, 0, 0, -1

Offset: 1

Gus Wiseman, Nov 20 2018

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).

			Triangle begins:
   1
   1
   1  -1
   1   1
   1  -1   1
   1   0  -1
   1   0  -1   1  -1
   1   2   1
   1   2  -1  -1   1
   1  -1   0   0   1
   1  -1   0   0   1  -1   1
   1   0   1  -1  -1
   1   0  -1   0   0   1   0   0  -1   1  -1
   1   0  -1   1   0   0  -1
For example, row 12 gives: p(211) = s(4) + s(31) - s(211) - s(1111).

Wikipedia, Symmetric polynomial

Row sums are A317554.

Cf. A000085, A008480, A056239, A082733, A124795, A153452, A296188, A296561, A300121, A304438, A317552, A317554, A321742-A321765.

A300121 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions connected skew partitions.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 4, 11, 12, 16, 12, 32, 28, 31, 8, 64, 31, 128, 33, 82, 64, 256, 28, 69, 144, 69, 86, 512, 105, 1024, 16, 208, 320, 209, 82, 2048, 704, 512, 86, 4096, 318, 8192, 216, 262, 1536, 16384, 64, 465, 262, 1232, 528, 32768, 209, 588, 245, 2912, 3328

Offset: 1

Gus Wiseman, Feb 25 2018

nonn

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns. A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(9) = 11 tableaux:
1 1
1 1
.
2 1   1 1   1 1   1 2
1 1   1 2   2 2   1 2
.
1 1   1 2   1 2   1 3
2 3   1 3   3 3   2 3
.
1 2   1 3
3 4   2 4

Cf. A000085, A000898, A056239, A006958, A138178, A153452, A238690, A259479, A259480, A296150, A296561, A297388, A299699, A299925, A299926, A300056, A300060, A300118, A300120, A300122, A300123, A300124.

undcon[y_]:=Select[Tuples[Range[0,#]&/@y],Function[v,GreaterEqual@@v&&With[{r=Select[Range[Length[y]],y[[#]]=!=v[[#]]&]},Or[Length[r]<=1,And@@Table[v[[i]]Table[PrimePi[p],{k}]]]];
Table[Length[cos[Reverse[primeMS[n]]]],{n,50}]

A321935 Tetrangle: T(n,H(u),H(v)) is the coefficient of p(v) in S(u), where u and v are integer partitions of n, H is Heinz number, p is the basis of power sum symmetric functions, and S is the basis of augmented Schur functions.

Original entry on oeis.org

1, 1, 1, -1, 1, 2, 3, 1, -1, 0, 1, 2, -3, 1, 6, 3, 8, 6, 1, 0, 3, -4, 0, 1, -2, -1, 0, 2, 1, 2, -1, 0, -2, 1, -6, 3, 8, -6, 1, 24, 30, 20, 15, 20, 10, 1, -6, 0, -5, 0, 5, 5, 1, 0, -6, 4, 3, -4, 2, 1, 0, 6, -4, 3, -4, -2, 1, 4, 0, 0, -5, 0, 0, 1, -6, 0, 5, 0, 5

Offset: 1

Gus Wiseman, Nov 23 2018

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

We define the augmented Schur functions to be S(y) = |y|! * s(y) / syt(y), where s is the basis of Schur functions and syt(y) is the number of standard Young tableaux of shape y.

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

Wikipedia, Symmetric polynomial

This is a regrouping of the triangle A321900.

Cf. A008480, A056239, A124794, A124795, A153452 (standard Young tableaux), A215366, A296188, A300121, A319191, A319193, A321908, A321912-A321934.

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cp's OEIS Frontend

A153734 Triangle T(n,k): T(n,k) gives the A153452(m_k) such that A056239(m_k) = n, [1<=k<=A000041(n)], sorted by m_k, read by rows. Sequence A060240 is this sequence's permutation.

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A153730 a(n) = Sum_{m=1..n} A153452(m).

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A154351 a(n) = number of distinct values of A056239(m) when A153452(m) is equal to n.

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A093641 Numbers of form 2^i * prime(j), i>=0, j>0, together with 1.

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A053529 a(n) = n! * number of partitions of n.

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A000898 a(n) = 2*(a(n-1) + (n-1)*a(n-2)) for n >= 2 with a(0) = 1.

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A296188 Number of normal semistandard Young tableaux whose shape is the integer partition with Heinz number n.

A000898 a(n) = 2(a(n-1) + (n-1)a(n-2)) for n >= 2 with a(0) = 1.