Real-rooted atlas

A225470 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 + 6\,x - x^{2} + 18\,n - 6\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-3 + 2\,x + 6\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
29211
399363138
413506309042074
522194121502295180536824
64286522630885953563003156757896
795265726382152164486721168014490504717780272
8239660208172224792490500367273476612360294847568468851120
9673536513651293898001582688160261783900256255651530900548101080913736439360
10209203521120167352424272551747295009963767880109599682576677678343035095040148510442870644160
11711799320816059410370432162071794709908402370646040486226050753845559564202658841705672015592519801115585873645440
12263318896863360228113319972096834961651215121724773620398422533770902501967872703941178220443448565202413552110244530257412594638564177280
131052399806697664094230572726384643598520419697112783989932117392109441376774730103787560536666890911217193240580543210748908172459457036936932761324450108421043200
14451935801295937280416816401584768768165272443536236400377256043751932805576525239685400567237406383648410503758414752148576538680816019454252227295070425675255405404095141077937296262937600
Technical details
Node id
n-776206235fcb89e5 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-0e58c6c76130f194

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          0;
  1:          1;
  2:          9,          2;
  3:         99,         36,          3;
  4:       1350,        630,         90,         4;
  5:      22194,      12150,       2295,       180,        5;
  6:     428652,     263088,      59535,      6300,      315,       6;
  7:    9526572,    6382152,    1644867,    211680,    14490,     504,     7;
  8:  239660208,  172224792,   49050036,   7273476,   612360,   29484,   756,    8;
  9: 6735365136, 5129389800, 1582688160, 261783900, 25625565, 1530900, 54810, 1080, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(6*n-6)*x+x^2)*P(n-2,x) + (6*n-3+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 18*n + 9 + (6*n - 6)*x + x^2)*P[n - 2, x] + (6*n - 3 + 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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (6*n - 6)*x + x^2)*v[i - 2] + (6*n - 3 + 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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.