A290030 -id:A290030 - cp's OEIS frontend

A290761 Irregular triangle, read by rows, of coefficients of polynomials that are the "nonstandard" factor of polynomials yielding the columns (up to sign) of triangle A290053, beginning with column 3.

3, 5, -6, 16, 1, 7, 16, 28, 0, 15, 225, 1265, 3707, 7120, 4900, -6480, 27648, 3, 83, 961, 6201, 24708, 60700, 87968, 85056, 0, 63, 2457, 41580, 404866, 2532971, 10651177, 30102338, 56577724, 72856616, 36562176, -51101568, 298598400, 9, 531, 14010, 219106

Offset: 1

Gregory Gerard Wojnar, Aug 09 2017

The polynomials come in pairs: first of odd degree; second of even degree 1 greater, whose constant term is always zero. Observations: All coefficients are positive except for the linear coefficients of the first polynomial in each pair, which are always negative. From the first of one pair to the first of the next pair, the degree always grows by 4. The "standard" factors of polynomials yielding the columns of triangle A290053 (beginning with column 3) are always of the form (1/A053657(k+2))*(N + k + 2) in odd rows of this triangle A290761, and of the form (N/A053657(k+2))*(N + k + 3)^2 in even rows of this triangle, where k is the row number. See examples.

			The first rows of the triangle are parsed as follows:
3, 5, -6, 16;
1, 7, 16, 28, 0;
15, 225, 1265, 3707, 7120, 4900, -6480, 27648;
3, 83, 961, 6201, 24708, 60700, 87968, 85056, 0;
63, 2457, 41580, 404866, 2532971, 10651177, 30102338, 56577724, 72856616, 36562176, -51101568, 298598400;
9, 531, 14010, 219106, 2266137, 16325259, 83797380, 307998768, 802828704, 1433652560, 1651979520, 1239918336, 0.
The associated full polynomials giving the columns of triangle A290053 are then:
(1/24) * (N + 3) * (3*N^3 + 5*N^2 - 6*N + 16);
(N/48) * (N + 5)^2 * (1*N^3 + 7*N^2 + 16*N + 28);
(1/5760) * (N + 5) * (15*N^7 + 225*N^6 + 1265*N^5 + 3707*N^4 + 7120*N^3 + 4900*N^2 - 6480*N + 27648);
(N/11520) * (N + 7)^2 * (3*N^7 + 83*N^6 + 961*N^5 + 6201*N^4 + 24708*N^3 + 60700*N^2 + 87968*N + 85056); etc.

G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017.

The first column of this triangle is A290030; alternating entries of the first column give A260326. See also triangle A290053, whose columns are A000012-A000096, A290061-A290071, A290127-A290723, etc.

A342645 Triangle read by rows: T(n,k) gives n! times the coefficient of x^k in the polynomial that describes the number of permutations on x letters with major index n.

Original entry on oeis.org

1, -1, 1, -2, -1, 1, 0, -7, 0, 1, 0, -14, -13, 2, 1, 120, -46, -65, -15, 5, 1, 0, 516, -356, -165, -5, 9, 1, 5040, 1392, 266, -1421, -280, 28, 14, 1, 0, 46320, 3772, -5740, -3871, -280, 98, 20, 1, 0, 215280, 212724, -26272, -31437, -7791, 126, 222, 27, 1

Offset: 0

Peter Kagey, Mar 17 2021

This n-th row describes a polynomial that eventually agrees with the n-th column of A008302.

Conjecture: For each m, T(n,n-m) is a polynomial of degree 2m whose leading coefficient is abs(A290030(m)/A053657(m+1)).

			n\k |    0       1       2       3       4      5    6    7   8  9
----+--------------------------------------------------------------
  0 |    1;
  1 |   -1,      1;
  2 |   -2,     -1,      1;
  3 |    0,     -7,      0,      1;
  4 |    0,    -14,    -13,      2,      1;
  5 |  120,    -46,    -65,    -15,      5,     1;
  6 |    0,    516,   -356,   -165,     -5,     9,   1;
  7 | 5040,   1392,    266,  -1421,   -280,    28,  14,   1;
  8 |    0,  46320,   3772,  -5740,  -3871,  -280,  98,  20,  1;
  9 |    0, 215280, 212724, -26272, -31437, -7791, 126, 222, 27, 1;
For n = 4, the polynomial that describes the 4th column of A008302 is
A008302(x,4) = (-14x -13x^2 +2x^3 + x^4)/4! = Sum_{j=0..4} (T(j,4)*x^j)/4!.

Peter Kagey, Rows n = 0..100, flattened
Mike Earnest, Does the number of permutations in S_n with major index equal to k, satisfy a degree k polynomial?, Mathematics Stack Exchange answer.

Cf. A008302, A053657, A290030.

A008302T[0, 0] := 1; A008302T[-1, k_] := 0;
A008302T[n_, k_] := (A008302T[n, k] = If[0 <= k <= n*(n - 1)/2, A008302T[n, k - 1] + A008302T[n - 1, k] - A008302T[n - 1, k - n], 0]);
A342645Row[n_] := (A342645Row[n] = Expand[n!*InterpolatingPolynomial[Table[{m, A008302T[m, n]}, {m, n, 2*n + 2}], x]]);
A342645T[n_, k_] := Coefficient[A342645Row[n], x, k];

Conjectures:
T(n,n)   = 1.
T(n,n-1) = (-3n + n^2)/2.
T(n,n-2) = (-2n + 21n^2 - 22n^3 + 3n^4)/24.
T(n,n-3) = (96n - 134n^2 + 13n^3 + 37n^4 - 13n^5 + n^6)/48.

cp's OEIS Frontend

A290761 Irregular triangle, read by rows, of coefficients of polynomials that are the "nonstandard" factor of polynomials yielding the columns (up to sign) of triangle A290053, beginning with column 3.

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