Real-rooted atlas

A269947 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-64 + 16\,x - x^{2} + 192\,n - 24\,n x - 240\,n^{2} + 12\,n^{2} x + 160\,n^{3} - 2\,n^{3} x - 60\,n^{4} + 12\,n^{5} - n^{6}\right)\,P_{n-2}(x) + \left(-9 + 2\,x + 15\,n - 9\,n^{2} + 2\,n^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1022
20235
301627447
4043275314451334
50276484884010220500687214
60345600061464721342620752751350711021724
70746496000133282195229820121617937675381930308782395841868

All 15 rows

Technical details
Node id
n-42247a28dbf8f7f6 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b56e8fc38add20bd

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 0,               2;
  2: 0,               2,               3;
  3: 0,              16,              27,              4;
  4: 0,             432,             753,            144,             5;
  5: 0,           27648,           48840,          10220,           500,           6;
  6: 0,         3456000,         6146472,        1342620,         75275,        1350,         7;
  7: 0,       746496000,      1332821952,      298201216,      17937675,      381930,      3087,       8;
  8: 0,    256048128000,    458277673536,   104060113024,    6525374045,   152527200,   1504426,    6272,     9;
  9: 0, 131096641536000, 235022241042432, 53889814766336, 3471066652320, 85924375254, 948214512, 4930608, 11664, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(n^6-12*n^5+60*n^4-160*n^3+240*n^2-192*n+64+(2*n^3-12*n^2+24*n-16)*x+x^2)*P(n-2,x) + (2*n^3-9*n^2+15*n-9+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64 + (2*n^3 - 12*n^2 + 24*n - 16)*x + x^2)*P[n - 2, x] + (2*n^3 - 9*n^2 + 15*n - 9 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64 + (2*n^3 - 12*n^2 + 24*n - 16)*x + x^2)*v[i - 2] + (2*n^3 - 9*n^2 + 15*n - 9 + 2*x)*v[i - 1]); 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

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.