A188181 -id:A188181 - cp's OEIS frontend

A188180 Number of strictly increasing arrangements of n numbers in -(n+6)..(n+6) with sum zero.

1, 8, 41, 207, 967, 4370, 19138, 81805, 343363, 1421530, 5820904, 23626234, 95220378, 381577586, 1521967986, 6047272980, 23951790860, 94618944122, 372969633775, 1467518524851, 5765549295661, 22623246705372, 88678543148354

Offset: 1

R. H. Hardin Mar 23 2011

nonn

Column 8 of A188181

			Some solutions for n=5
.-5...-7...-6...-9..-11...-8...-9...-8...-9..-10...-6..-10...-6...-8...-6...-8
.-2...-6...-3...-5...-3...-7...-2...-7...-6...-7...-5...-4...-4...-7...-1...-1
.-1...-3....1....0...-1....4....2...-2...-5....0...-2....2...-2....0....0....0
..3....5....3....6....7....5....3....7....9....6....3....4....4....4....1....1
..5...11....5....8....8....6....6...10...11...11...10....8....8...11....6....8

R. H. Hardin, Table of n, a(n) for n = 1..25

A188182 Number of strictly increasing arrangements of 4 numbers in -(n+2)..(n+2) with sum zero.

Original entry on oeis.org

5, 12, 24, 43, 69, 104, 150, 207, 277, 362, 462, 579, 715, 870, 1046, 1245, 1467, 1714, 1988, 2289, 2619, 2980, 3372, 3797, 4257, 4752, 5284, 5855, 6465, 7116, 7810, 8547, 9329, 10158, 11034, 11959, 12935, 13962, 15042, 16177, 17367, 18614, 19920, 21285

Offset: 1

R. H. Hardin Mar 23 2011

nonn

Row 4 of A188181

			Some solutions for n=5
.-6...-7...-6...-5...-3...-7...-4...-5...-4...-5...-6...-2...-6...-6...-7...-4
.-4....0...-1...-4....0...-1...-3...-3...-2...-2...-5...-1...-3...-3....0...-3
..3....1....3....4....1....2....0....1....2....2....5....0....3....2....3....1
..7....6....4....5....2....6....7....7....4....5....6....3....6....7....4....6

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n)=3*a(n-1)-3*a(n-2)+2*a(n-3)-3*a(n-4)+3*a(n-5)-a(n-6).

Empirical: a(n) = (n+1)*(4*n^2+17*n+22)/18 -2 *A049347(n)/9; g.f. -x*(-5+3*x-3*x^2+3*x^3-3*x^4+x^5) / ( (1+x+x^2)*(x-1)^4 ). - R. J. Mathar, Mar 26 2011

A188184 Number of strictly increasing arrangements of 6 numbers in -(n+4)..(n+4) with sum zero.

Original entry on oeis.org

32, 94, 227, 480, 920, 1636, 2739, 4370, 6698, 9926, 14293, 20076, 27594, 37212, 49341, 64444, 83036, 105690, 133037, 165772, 204654, 250510, 304239, 366814, 439284, 522780, 618513, 727782, 851974, 992568, 1151137, 1329352, 1528984

Offset: 1

R. H. Hardin Mar 23 2011

nonn

Row 6 of A188181

			Some solutions for n=5
.-7...-9...-6...-4...-8...-7...-7...-6...-7...-6...-8...-9...-6...-6...-9...-8
.-5...-5...-4...-2...-6...-5...-2...-5...-3...-3...-3...-2...-3...-5...-6...-3
.-4....1...-1...-1....1...-4...-1...-2....0...-2...-2....0...-2...-4....1...-1
..2....3....2....1....2....1....0....0....1...-1...-1....2....1....0....3....1
..5....4....4....2....3....7....3....6....3....5....5....4....3....7....4....3
..9....6....5....4....8....8....7....7....6....7....9....5....7....8....7....8

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n)=3*a(n-1)-2*a(n-2)-a(n-3)+2*a(n-5)-a(n-6)-a(n-7)+2*a(n-8)-a(n-10)-2*a(n-11)+3*a(n-12)-a(n-13).

Empirical: G.f. -x*(-32 +2*x -9*x^2 -19*x^3 -28*x^4 +x^5 +5*x^6 -17*x^7 +x^8 +10*x^9 +13*x^10 -23*x^11 +8*x^12) / ( (1+x) *(1+x+x^2) *(x^4+x^3+x^2+x+1) *(x-1)^6 ). - R. J. Mathar, Mar 26 2011

A188186 Number of strictly increasing arrangements of 8 numbers in -(n+6)..(n+6) with sum zero.

Original entry on oeis.org

289, 910, 2430, 5744, 12346, 24591, 46029, 81805, 139143, 227930, 361384, 556834, 836618, 1229093, 1769773, 2502617, 3481445, 4771508, 6451232, 8614108, 11370764, 14851235, 19207395, 24615603, 31279561, 39433366, 49344790, 61318804, 75701312

Offset: 1

R. H. Hardin Mar 23 2011

nonn

Row 8 of A188181

			Some solutions for n=5
.-9..-10..-10...-8..-11...-8...-9..-11...-9..-11..-11...-9..-11..-10..-10...-9
.-8...-8...-9...-7...-3...-5...-8..-10...-8...-7...-5...-7...-4...-9...-9...-7
.-5...-4...-7...-6...-2...-4...-5...-2...-6...-6...-4...-5...-3...-7...-7...-4
.-1....1....0...-2...-1...-3...-4....1...-1...-3...-3...-2...-2...-1....0...-1
..1....2....2....0....1....0...-2....2....3....2....0....0....2....4....4....0
..3....5....6....5....3....5....7....3....4....6....4....3....5....5....5....2
..9....6....8....7....5....6...10....6....7....9....8....9....6....7....6....9
.10....8...10...11....8....9...11...11...10...10...11...11....7...11...11...10

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n)=3*a(n-1)-2*a(n-2)-3*a(n-4)+4*a(n-5)-3*a(n-8)+3*a(n-9)-a(n-11)-a(n-12)+3*a(n-14)-3*a(n-15)+4*a(n-18)-3*a(n-19)-2*a(n-21)+3*a(n-22)-a(n-23).

Empirical: G.f. -x*(-289 -43*x -278*x^2 -274*x^3 -841*x^4 -615*x^5 -598*x^6 -412*x^7 -715*x^8 -363*x^9 -163*x^10 -72*x^11 -98*x^12 -200*x^13 +217*x^14 -5*x^15 -49*x^16 -253*x^17 +221*x^18 +23*x^19 +108*x^20 -206*x^21 +73*x^22) / ( (1+x) *(x^4+x^3+x^2+x+1) *(x^6+x^5+x^4+x^3+x^2+x+1) *(1+x+x^2)^2 *(x-1)^8 ). - R. J. Mathar, Mar 26 2011

A331545 Triangle of constant term of the symmetric q-binomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 3, 0, 3, 0, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 0, 4, 0, 8, 0, 4, 0, 1, 1, 1, 4, 8, 12, 12, 8, 4, 1, 1, 1, 0, 5, 0, 18, 0, 18, 0, 5, 0, 1, 1, 1, 5, 13, 24, 32, 32, 24, 13, 5, 1, 1, 1, 0, 6, 0, 33

Offset: 0

Michael Somos, Jan 19 2020

nonn

Symmetric q-binomial coefficients are based on symmetric q-numbers [n] := (q^n-1/q^n)/(q-1/q).

			Triangle begins:
  n\k| 0 1 2 3 4 5 6 7  ...
  ---+----------------
   0 | 1
   1 | 1 1
   2 | 1 0 1
   3 | 1 1 1 1
   4 | 1 0 2 0 1
   5 | 1 1 2 2 1 1
   6 | 1 0 3 0 3 0 1
   7 | 1 1 3 5 5 3 1 1
   ...

CF. A000982, A001973, A063074, A188181.

T[ n_, k_] := Coefficient[ QBinomial[ n, k, x^2] / x^(k (n - k)) // FunctionExpand // Expand, x, 0];

{T(n, k) = if( k<0 || k>n, 0, polcoeff( prod(j = 1, k, (x^(n+1-j) - x^(-n-1+j))/(x^j - x^(-j))), 0))};

T(2*n, 2*k+1) = 0. T(2*n+1, 3) = A000982(n). T(2*n+1, 5) = A001973(n) if n>=2. T(4*n, 2*n) = A063074(n).

cp's OEIS Frontend

A188180 Number of strictly increasing arrangements of n numbers in -(n+6)..(n+6) with sum zero.

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A188182 Number of strictly increasing arrangements of 4 numbers in -(n+2)..(n+2) with sum zero.

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A188184 Number of strictly increasing arrangements of 6 numbers in -(n+4)..(n+4) with sum zero.

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A188186 Number of strictly increasing arrangements of 8 numbers in -(n+6)..(n+6) with sum zero.

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A331545 Triangle of constant term of the symmetric q-binomial coefficients.

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