Real-rooted atlas

A034870 → D → D

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 - 5\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-n - 2\,n x - n x^{2} + 2\,n^{2} + 4\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
00
122
212241248
33012018012030480
4563368401120840336563584
590720252050406300504025207209023040
613213205940158402772033264277201584059401320132135168
718221841201240040900901441441681681441449009040040120122184182745472

All 15 rows

Technical details
Node id
n-95f05612a47cc246 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2cd9f46ca2d53115

Observed on the stored rows ?

  • Degrees: -1, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26 (affine: slope 2, intercept -2)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   0;
  1:   2;
  2:  12,   24,    12;
  3:  30,  120,   180,    120,     30;
  4:  56,  336,   840,   1120,    840,     336,      56;
  5:  90,  720,  2520,   5040,   6300,    5040,    2520,     720,      90;
  6: 132, 1320,  5940,  15840,  27720,   33264,   27720,   15840,    5940,    1320,     132;
  7: 182, 2184, 12012,  40040,  90090,  144144,  168168,  144144,   90090,   40040,   12012,    2184,    182;
  8: 240, 3360, 21840,  87360, 240240,  480480,  720720,  823680,  720720,  480480,  240240,   87360,  21840,   3360,   240;
  9: 306, 4896, 36720, 171360, 556920, 1336608, 2450448, 3500640, 3938220, 3500640, 2450448, 1336608, 556920, 171360, 36720, 4896, 306;
%F Formula
The row polynomials satisfy (2*n^2-5*n+3)*P(n,x) = (2*n^2-n+(4*n^2-2*n)*x+(2*n^2-n)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - n + (4*n^2 - 2*n)*x + (2*n^2 - n)*x^2)*P[n - 1, x])/(2*n^2 - 5*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 - n + (4*n^2 - 2*n)*x + (2*n^2 - n)*x^2)*v[i - 1])/(2*n^2 - 5*n + 3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.