Real-rooted atlas

A257609 → D (shift +2) → reverse

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-2 - 2\,x^{2} - 3\,n - 2\,n x - 3\,n x^{2} - n^{2} - 3\,n^{2} x - n^{2} x^{2} - n^{3} x\right)\,P_{n-2}(x) + \left(-2\,x + 2\,x^{3} - 3\,n x + 2\,n x^{2} + n x^{3} - n^{2} x + n^{2} x^{2}\right)\,P_{n-2}'(x) + \left(2 + 4\,x + 3\,n + 4\,n x + n^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(x - x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 3 + 22\,x + 11\,x^{2}\)
\(P_{2}(x) = 4 + 78\,x + 132\,x^{2} + 26\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
13221136
247813226240
35228906604571800
4660047647248238212015120
5714822146562476468578586247141120
68351487648441170624760264702292165021451520
798104334880273115265517705241416136557695680101316329600
8101832412210961542441658428684786212403895245666104643052742036199584000

All 15 rows

Technical details
Node id
n-6e6daac86c9463fd · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-b00b790aa67df404

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  3,    22,       11;
   2:  4,    78,      132,        26;
   3:  5,   228,      906,       604,         57;
   4:  6,   600,     4764,      7248,       2382,         120;
   5:  7,  1482,    21465,     62476,      46857,        8586,         247;
   6:  8,  3514,    87648,    441170,     624760,      264702,       29216,        502;
   7:  9,  8104,   334880,   2731152,    6551770,     5241416,     1365576,      95680,       1013;
   8: 10, 18324,  1221096,  15424416,   58428684,    78621240,    38952456,    6610464,     305274,      2036;
   9: 11, 40830,  4304439,  81501480,  464229318,   975073716,   812561430,  265273896,   30563055,    956542,    4083;
  10: 12, 89958, 14797260, 409801050, 3386252280, 10539350556, 13651032024, 7528107540, 1693126140, 136600350, 2959452, 8178;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(n^2+3*n+2+(n^3+3*n^2+2*n)*x+(n^2+3*n+2)*x^2)*P(n-2,x) - ((n^2+3*n+2)*x-(n^2+2*n)*x^2-(n+2)*x^3)*d/dx P(n-2,x) + (n^2+3*n+2+(n^2+4*n+4)*x)*P(n-1,x) + ((n+1)*x-(n+1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 3 + 22*x + 11*x^2, P(2,x) = 4 + 78*x + 132*x^2 + 26*x^3.
%t Mathematica
P[1, x_] := 3 + 22*x + 11*x^2;
P[2, x_] := 4 + 78*x + 132*x^2 + 26*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 3*n + 2 + (n^3 + 3*n^2 + 2*n)*x + (n^2 + 3*n + 2)*x^2)*P[n - 2, x] - ((n^2 + 3*n + 2)*x - (n^2 + 2*n)*x^2 - (n + 2)*x^3)*D[P[n - 2, x], x] + (n^2 + 3*n + 2 + (n^2 + 4*n + 4)*x)*P[n - 1, x] + ((n + 1)*x - (n + 1)*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 3 + 22*x + 11*x^2; v[2] = 4 + 78*x + 132*x^2 + 26*x^3; for(n = 3, N, my(i = n); v[i] = (-(n^2 + 3*n + 2 + (n^3 + 3*n^2 + 2*n)*x + (n^2 + 3*n + 2)*x^2)*v[i - 2] - ((n^2 + 3*n + 2)*x - (n^2 + 2*n)*x^2 - (n + 2)*x^3)*deriv(v[i - 2], x) + (n^2 + 3*n + 2 + (n^2 + 4*n + 4)*x)*v[i - 1] + ((n + 1)*x - (n + 1)*x^2)*deriv(v[i - 1], x))/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.