A076822 -id:A076822 - cp's OEIS frontend

A188181 T(n,k) is the number of strictly increasing arrangements of n numbers in -(n+k-2)..(n+k-2) with sum zero.

1, 1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 4, 8, 12, 12, 1, 5, 13, 24, 32, 32, 1, 6, 18, 43, 73, 94, 94, 1, 7, 25, 69, 141, 227, 289, 289, 1, 8, 32, 104, 252, 480, 734, 910, 910, 1, 9, 41, 150, 414, 920, 1656, 2430, 2934, 2934, 1, 10, 50, 207, 649, 1636, 3370, 5744, 8150, 9686, 9686, 1, 11

Offset: 1

R. H. Hardin, Mar 23 2011

			Table starts
....1....1.....1.....1......1......1......1.......1.......1.......1.......1
....1....2.....3.....4......5......6......7.......8.......9......10......11
....2....5.....8....13.....18.....25.....32......41......50......61......72
....5...12....24....43.....69....104....150.....207.....277.....362.....462
...12...32....73...141....252....414....649.....967....1394....1944....2649
...32...94...227...480....920...1636...2739....4370....6698....9926...14293
...94..289...734..1656...3370...6375..11322...19138...30982...48417...73316
..289..910..2430..5744..12346..24591..46029...81805..139143..227930..361384
..910.2934..8150.20094..45207..94257.184717..343363..610358.1043534.1724882
.2934.9686.27718.70922.165821.360002.734517.1421530.2628824.4672836.8022362
Some solutions for n=7 and k=5:
.-7...-9...-8..-10...-6...-6...-9...-8...-8...-7...-9..-10...-9...-8...-9...-7
.-5...-7...-6...-7...-5...-5...-4...-6...-7...-3...-8...-5...-3...-7...-5...-4
.-3...-1...-5...-4...-4...-3...-1...-4...-5...-2...-6...-3...-1...-3...-4...-3
.-1....0...-1....3...-3...-2....0...-1....0...-1....4....0....1....1...-1...-1
..3....3....2....5...-1....1....1....4....3....0....5....3....3....2....4....0
..4....4....8....6....9....7....4....5....7....4....6....5....4....5....6....7
..9...10...10....7...10....8....9...10...10....9....8...10....5...10....9....8

R. H. Hardin, Table of n, a(n) for n = 1..550

Column 1 is A076822.
Column 2 is A002838.
Cf. A000982.

T(3,n) = A000982(n+1).

A002838 Balancing weights on the integer line.

Original entry on oeis.org

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: 1

N. J. A. Sloane

Also number of partitions of n(n+1)/2 into up to n parts each no greater than n+1, partitions of n(n+3)/2 into exactly n parts each no greater than n+2 and partitions of n(n+1) into exactly n distinct parts each no greater than 2n+1, thus providing balancing solutions for n weights in distinct integer positions on [ -n,n] with a pivot at 0. - Henry Bottomley, Aug 09 2002

Is this a shifted version of A076822? - Vladimir Reshetnikov, Oct 06 2016

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).

R. E. Odeh and E. J. Cockayne, Balancing weights on the integer line, J. Combin. Theory, 7 (1969), 130-135.

Cf. A047997, A076822, A188181 (columns 1, 2).

(* This program is not convenient for large values of n *) a[n_] := Length[ IntegerPartitions[n*(n+1)/2, n, Range[n+1]]]; Table[ Print[{n, an = a[n]}]; an, {n, 1, 16}] (* Jean-François Alcover, Jan 02 2013 *)

a(n) = A047997(n, n) = A067059(n, n+1). a(n) tends towards (sqrt(12)/Pi)*4^n/n^2 and something like (sqrt(12)/Pi)*4^n/(n^2+1.85*n+0.8) seems to give an even closer approximation. - Henry Bottomley, Aug 09 2002

More terms from Henry Bottomley, Aug 09 2002

A277218 Maximal coefficient among the polynomials in row n of the triangle of q-binomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 5, 8, 12, 20, 32, 58, 94, 169, 289, 526, 910, 1667, 2934, 5448, 9686, 18084, 32540, 61108, 110780, 208960, 381676, 723354, 1328980, 2527074, 4669367, 8908546, 16535154, 31630390, 58965214, 113093022, 211591218, 406680465, 763535450, 1470597342, 2769176514, 5342750699, 10089240974

Offset: 0

Vladimir Reshetnikov, Oct 05 2016

nonn

q-binomial coefficients are polynomials in q with integer coefficients.

Is A055606 a shifted version of this sequence?

			Row 5 of the triangle of q-binomial coefficients is [1, 1 + q + q^2 + q^3 + q^4, 1 + q + 2*q^2 + 2*q^3 + 2*q^4 + q^5 + q^6, 1 + q + 2*q^2 + 2*q^3 + 2*q^4 + q^5 + q^6, 1 + q + q^2 + q^3 + q^4, 1], so the max coefficient is 2. Hence a(5) = 2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..200
E. Friedman and M. Keith, Magic Carpets, J. Int Sequences, 3 (2000), #P.00.2.5.
Eric W. Weisstein, q-Binomial Coefficient
Wikipedia, q-binomial

Cf. A002838, A022166, A029895, A055606, A076822.

f:= proc(n) local k, c, v, q;
  uses QDifferenceEquations;
  v:= 0:
  for k from 0 to n do
    c:= coeffs(expand(expand(QBinomial(n,k,q))),q);
    v:= max(v, max(c));
  od:
v
end proc:
map(f, [$0..50]); # Robert Israel, Oct 05 2016

Table[Coefficient[Expand[FunctionExpand[QBinomial[n, Floor[n/2], q]]], q, Floor[n^2/8]], {n, 0, 30}] (* Vladimir Reshetnikov, Sep 24 2021 *)

a(n) ~ sqrt(3) * 2^(n+2) / (Pi * n^2). - Vaclav Kotesovec, Oct 09 2016

A119551 Number of ordered finite sequences a_1 <= a_2 <= ... <= a_n of length n of positive integers less than or equal to n whose product is n! and whose sum is n * (n + 1) / 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 6, 6, 22, 22, 60, 159, 377, 377, 1007, 1007, 2867, 8147, 22403, 22403, 67808, 176128, 495053, 1362240, 4210266, 4210266, 14223808, 14223808, 42235255, 129279396, 370630653, 1178215490

Offset: 0

Jens Voß, May 30 2006

a(n) is also the number of lattice points in a sequence of polytopes. Given n, define a vector x(k) = #{j : a_j = k} and define a matrix A with n columns as follows: first row all 1 (gives length of a_j); second row 1,2,...,n (sum of a_j); finally one row for each prime p <= n with entries A(row p, column k) = maximum exponent of p that divides k, e.g., A(p=2,k=8)=3 because 2^3|8 (this gives factorization of product of a_j). Then a(n) is the number of nonnegative integer lattice points in the polytope A*x = A*(1,1,1...)T. - Martin Fuller, Feb 12 2023

			a(9) = 2 because the sequences (1, 2, 3, 4, 5, 6, 7, 8, 9) and (1, 2, 4, 4, 4, 5, 7, 9, 9) both add up to 45 and multiply up to 9!.

Martin Fuller, Table of n, a(n) for n = 0..61

Cf. A000040, A000142, A000217, A076822 without restriction on product, A120690 without restriction on sum.

a[n_] := a[n] = Module[{b}, b[c_, s_, p_, m_] := b[c, s, p, m] = Module[{x}, If[c <= 0 || m <= 1 || s <= c || s > m*c, Boole[ c == s && p == 1], x = IntegerExponent[p, m]; Sum[b[c - i, s - m*i, p/m^i, m - 1], {i, x*Boole@PrimeQ[m], x} ]]]; b[n, n*(n + 1)/2, n!, n]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 32}] (* Jean-François Alcover, Jul 05 2022, after Martin Fuller *)

a(n) = (b(c,s,p,m) = local(x); if(c<=0||m<=1||s<=c||s>m*c, c==s&&p==1, x=valuation(p,m); sum(i=x*isprime(m), x, b(c-i,s-m*i,p/m^i,m-1)))); b(n,n*(n+1)/2,n!,n) \\ Martin Fuller, Jun 26 2006

a(p) = a(p-1) for prime p. - Alois P. Heinz, Jul 05 2022

a(18) and a(19) from John W. Layman, Jun 08 2006
More terms from Martin Fuller, Jun 26 2006
a(0)=1 prepended by Alois P. Heinz, Jul 05 2022
a(36)-a(61) from Martin Fuller, Feb 12 2023

A277271 Second largest coefficient among the polynomials in row n of the triangle of q-binomial coefficients.

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 19, 30, 55, 90, 166, 285, 519, 902, 1656, 2929, 5424, 9673, 18012, 32467, 60981, 110599, 208445, 381301, 722552, 1327869, 2522994, 4665786, 8902311, 16524759, 31594853, 58935171, 113038371, 211499060, 406350261, 763246536, 1470080699

Offset: 4

Vladimir Reshetnikov, Oct 07 2016

nonn

q-binomial coefficients are polynomials in q with integer coefficients.

			Row 5 of the triangle of q-binomial coefficients is [1, 1 + q + q^2 + q^3 + q^4, 1 + q + 2*q^2 + 2*q^3 + 2*q^4 + q^5 + q^6, 1 + q + 2*q^2 + 2*q^3 + 2*q^4 + q^5 + q^6, 1 + q + q^2 + q^3 + q^4, 1]. The largest coefficient is 2, and the second largest coefficient is 1. Hence A277218(5) = 2 and a(5) = 1.

Eric W. Weisstein, q-Binomial Coefficient
Wikipedia, q-binomial

Cf. A002838, A022166, A029895, A055606, A076822, A277218 (largest coefficients).

Table[(Union @@ Table[CoefficientList[FunctionExpand[QBinomial[n, k, q]], q], {k, 0, n}])[[-2]], {n, 4, 40}]

cp's OEIS Frontend

A188181 T(n,k) is the number of strictly increasing arrangements of n numbers in -(n+k-2)..(n+k-2) with sum zero.

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A002838 Balancing weights on the integer line.

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A277218 Maximal coefficient among the polynomials in row n of the triangle of q-binomial coefficients.

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A119551 Number of ordered finite sequences a_1 <= a_2 <= ... <= a_n of length n of positive integers less than or equal to n whose product is n! and whose sum is n * (n + 1) / 2.

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A277271 Second largest coefficient among the polynomials in row n of the triangle of q-binomial coefficients.

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Mathematica