A039744 -id:A039744 - cp's OEIS frontend

A188211 T(n,k)=Number of nondecreasing arrangements of n numbers in -(n+k-2)..(n+k-2) with sum zero.

1, 1, 2, 1, 3, 5, 1, 4, 8, 18, 1, 5, 13, 33, 73, 1, 6, 18, 55, 141, 338, 1, 7, 25, 86, 252, 676, 1656, 1, 8, 32, 126, 414, 1242, 3370, 8512, 1, 9, 41, 177, 649, 2137, 6375, 17575, 45207, 1, 10, 50, 241, 967, 3486, 11322, 33885, 94257, 246448, 1, 11, 61, 318, 1394, 5444

Offset: 1

R. H. Hardin Mar 24 2011

Table starts
......1......1.......1.......1.......1.......1........1........1........1
......2......3.......4.......5.......6.......7........8........9.......10
......5......8......13......18......25......32.......41.......50.......61
.....18.....33......55......86.....126.....177......241......318......410
.....73....141.....252.....414.....649.....967.....1394.....1944.....2649
....338....676....1242....2137....3486....5444.....8196....11963....17002
...1656...3370....6375...11322...19138...30982....48417....73316...108108
...8512..17575...33885...61731..107233..178870...288100...450096...684572
..45207..94257..184717..343363..610358.1043534..1724882..2767118..4323349
.246448.517971.1028172.1943488.3521260.6147894.10388788.17052653.27273240

			Some solutions for n=5 k=3
.-5...-5...-4...-4...-6...-3...-6...-4...-2...-6...-2...-6...-4...-5...-5...-4
.-2...-1...-4...-2...-5...-3...-1...-3...-1...-6...-1...-1...-3...-4...-3...-4
..0....0...-1...-2...-1...-1....0...-1...-1....1...-1....0....1...-1....0....1
..1....0....3....3....6....2....2....3....1....5....0....1....1....5....4....3
..6....6....6....5....6....5....5....5....3....6....4....6....5....5....4....4

R. H. Hardin, Table of n, a(n) for n = 1..430

Column 1 is A039744
Column 2 is A109655
Row 3 is A000982(n+2)
Row 5 is A188183(n+2)
Row 7 is A188185(n+3)

A007878 Number of terms in discriminant of generic polynomial of degree n.

Original entry on oeis.org

1, 2, 5, 16, 59, 246, 1103, 5247, 26059, 133881, 706799, 3815311, 20979619, 117178725, 663316190, 3798697446, 21976689397

Offset: 1

reiner(AT)math.umn.edu

Here "generic" means that the coefficients are algebraically independent symbols. - Robert Israel, Oct 02 2015
At one point it was suggested that this is the same sequence as A039744, but this is wrong. Dean Hickerson, Dec 16 2006, comments as follows: (Start)
The claim that A039744 equals the number of monomials in the discriminant is false. The first counterexample is n=4: There are 18 such partitions, but the discriminant has no terms corresponding to the partitions 3+2+2+2+2+1 and 2+2+2+2+2+2, so the number of monomials in the discriminant is only 16.
Columns near the left or right have very few nonzero elements and this adds some restrictions to the partitions.
For example, from column 2 of the matrix, we see that the partition must have at least one term equal to n or n-1. From the last column, it must have at least one term equal to 0 or 1. Maybe the complete list of such conditions is enough; I don't know.
Even if we could figure out exactly which partitions correspond to monomials that occur in the expansion, I can't rule out the possibility that the coefficients of some such monomial could cancel out, further reducing the number of nonzero monomials in the discriminant. (End)

			Discriminant of a_0 + a_1 x + ... + a_n x^n is 1/a_n times the determinant of a particular matrix; for n=4 that matrix is
  [ a_4   a_3   a_2   a_1   a_0   0     0    ]
  [ 0     a_4   a_3   a_2   a_1   a_0   0    ]
  [ 0     0     a_4   a_3   a_2   a_1   a_0  ]
  [ 4a_4  3a_3  2a_2  1a_1  0     0     0    ]
  [ 0     4a_4  3a_3  2a_2  1a_1  0     0    ]
  [ 0     0     4a_4  3a_3  2a_2  1a_1  0    ]
  [ 0     0     0     4a_4  3a_3  2a_2  1a_1 ]
It is easy to see that there are no monomials in the expansion of this involving either a_4 * a_3 * a_2^4 * a_1 or a_4 * a_2^6.
The discriminant of the cubic K3*x^3 + K2*x^2 + K1*x + K0 is -27*K3^2*K0^2 + 18*K3*K2*K1*K0 - 4*K2^3*K0 - 4*K3*K1^3 + K2^2*K1^2 which contains 5 monomials. - Bill Daly (bill.daly(AT)tradition.co.uk)

Kinji Kimura, Computing the general discriminant formula of degree 17, presented at Gröbner Bases, Resultants and Linear Algebra workshop, September 3-6, 2013, Research Institute for Symbolic Computation, Hagenberg, Austria. See abstract and author's website.
Kinji Kimura, programs written in Maple, Magma, etc.

function Disc(n) F := FunctionField(Rationals(),n); R := PolynomialRing(F); f := x^n + &+[ (F.i)*x^(n-i) : i in [ 1..n ] ]; return Discriminant(f); end function; [ #Monomials(Numerator(Disc(n))) : n in [ 1..7 ] ] // Victor S. Miller, Dec 16 2006

A007878 := proc(n) local x,a,ii; nops(discrim(sum(a[ ii ]*x^ii, ii=0..n), x)) end;

Clear[f, g]; g[0] = f[0]; g[n_Integer?Positive] := g[n] = g[n - 1] + f[n] x^n; myFun[n_Integer?Positive] := Length@Resultant[g[n], D[g[n], x], x, Method -> "BezoutMatrix"]; Table[myFun[n], {n, 1, 8}] (* Artur Jasinski, improved by Jean-Marc Gulliet (jeanmarc.gulliet(AT)gmail.com) *)

A = InfinitePolynomialRing(QQ, 'a')
a = A.gen()
for N in range(1, 7):
    x = polygen(A, 'x')
    P = sum(a[i] * x^i for i in range(N + 1))
    M = P.sylvester_matrix(diff(P, x), x)
    print(M.determinant().number_of_terms())
# Georg Muntingh, Jan 17 2014

a(9) from Lyle Ramshaw (ramshaw(AT)pa.dec.com)
Entry revised by N. J. A. Sloane, Dec 16 2006
a(10) from Artur Jasinski, Apr 02 2008
a(11) from Georg Muntingh, Jan 17 2014
a(12) from Georg Muntingh, Mar 10 2014
a(13)-a(14) from Seiichi Manyama, Nov 08 2023
a(15)-a(17) from Kimura (2013) added by Andrey Zabolotskiy, Jun 30 2024

A076822 Number of partitions of the n-th triangular number involving only the numbers 1..n and with exactly n terms.

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 32, 94, 289, 910, 2934, 9686, 32540, 110780, 381676, 1328980, 4669367, 16535154, 58965214, 211591218, 763535450, 2769176514, 10089240974, 36912710568, 135565151486, 499619269774, 1847267563742, 6850369296298

Offset: 0

Jon Perry, Nov 19 2002

nonn

Asymptotic to (sqrt(3)/(2*Pi))*(4^n/n^2). It is the number of lattice paths from (0,0) to (n,n-1) with steps only to the right or upward and having area n(n-1)/2 between the path and the x-axis. In the reference by Takács use formula (77) with a=n, b=n(n-1)/2 and then Stirling's formula. - Kent E. Morrison, May 28 2016

a(n) is the number of fair dice with n sides and expected value (n+1)/2 with distinct composition of numbers between 1 and n. - Felix Huber, Aug 02 2024

			a(4)=5 as T(4)=10= 1+1+4+4 =1+2+3+4 = 1+3+3+3 = 2+2+2+4 = 2+2+3+3.

Max Alekseyev and Alois P. Heinz, Table of n, a(n) for n = 0..240 (terms n=1..100 from Max Alekseyev)
L. Takács, Some asymptotic formulas for lattice paths, J. Statist. Plann. Inference, 14 (1986), 123-142.

Cf. A067059, A047993, A039744.
Cf. A002838. [From R. J. Mathar, Sep 20 2008]
Cf. A188181 (columns 1, 2).

ccc=new Array(); cccc=0;
for (n=1; n<11; n++)
{
    str='cc=0; for (i1=1; i1<'+(n+1)+'; i1++)';
    str2='i1';
    str3='i1';
    tn=1;
    for (i=2; i<=n; i++)
    {
        str+='for (i'+i+'=i'+(i-1)+'; i'+i+'<'+(n+1)+'; i'+i+'++)';
        str2+='+i'+i;
        str3+=', ", ", i'+i;
        tn+=i;
    }
    str+='if ('+str2+'=='+tn+') document.print(++cc, ":", '+str3+', "
")';
    eval(str);
    ccc[cccc++ ]=cc;
    document.print('****
');
}
document.write(ccc);

f[n_] := Block[{p = IntegerPartitions[n(n + 1)/2, n]}, Length[ Select[p, Length[ # ] == n &]]]; Table[ f[n], {n, 1, 13}]

a(n) = A067059(n,n+1); also a(n) = T[n*(n-1)/2, n-1, n] with T[ ] defined as in A047993. - Martin Fuller, Jun 27 2006

Edited and extended to 12 terms by Robert G. Wilson v, Nov 23 2002
Further terms from Max Alekseyev, May 24 2007
a(0)=1 prepended by Alois P. Heinz, May 28 2016

A362967 Number of nondecreasing partitions of n^2 into n parts from the intervals [1,n], [2,n+1], ..., [n,2n-1], respectively.

Original entry on oeis.org

1, 1, 2, 5, 17, 66, 294, 1393, 6965, 36111, 192850, 1053894, 5871306, 33234990, 190704140, 1107086841, 6492325565, 38412063755, 229052915708, 1375396927729, 8310509070747, 50496841617102, 308394124109340, 1892137615326526, 11658149626059204, 72108088307308032, 447590893613564372

Offset: 0

Max Alekseyev, Jun 16 2023

nonn

a(n) <= A039744(n); a(n) <= A077045(n); a(n) >= A019589(n).

Max Alekseyev, Table of n, a(n) for n = 0..100

Cf. A019589, A039744, A077045.

def a362967(n): return Partitions(n^2, length=n, inner=range(n,0,-1), outer=range(2*n-1,n-1,-1)).cardinality()

cp's OEIS Frontend

A188211 T(n,k)=Number of nondecreasing arrangements of n numbers in -(n+k-2)..(n+k-2) with sum zero.

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A007878 Number of terms in discriminant of generic polynomial of degree n.

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Magma

Maple

Mathematica

Sage

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A076822 Number of partitions of the n-th triangular number involving only the numbers 1..n and with exactly n terms.

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Comments

Examples

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Crossrefs

Programs

JavaScript

Mathematica

Formula

Extensions

A362967 Number of nondecreasing partitions of n^2 into n parts from the intervals [1,n], [2,n+1], ..., [n,2n-1], respectively.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Sage