Real-rooted atlas

A034870 → basis A013609 → D

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(4\,n + 8\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 8 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
18816
26419219264512
3384192038403840192038412288
420481433643008716807168043008143362048262144
510240921603686408601601290240129024086016036864092160102405242880
64915254067227033608110080162201602270822422708224162201608110080270336054067249152100663296
722937629818881789132865601536164003840295206912393609216393609216295206912164003840656015361789132829818882293761879048192

All 15 rows

Technical details
Node id
n-70753dc7278bcc95 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f001db5d15224ada

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, intercept -1)
  • 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:       8,        8;
  2:      64,      192,       192,         64;
  3:     384,     1920,      3840,       3840,        1920,         384;
  4:    2048,    14336,     43008,      71680,       71680,       43008,       14336,        2048;
  5:   10240,    92160,    368640,     860160,     1290240,     1290240,      860160,      368640,        92160,        10240;
  6:   49152,   540672,   2703360,    8110080,    16220160,    22708224,    22708224,    16220160,      8110080,      2703360,      540672,       49152;
  7:  229376,  2981888,  17891328,   65601536,   164003840,   295206912,   393609216,   393609216,    295206912,    164003840,    65601536,    17891328,     2981888,      229376;
  8: 1048576, 15728640, 110100480,  477102080,  1431306240,  3148873728,  5248122880,  6747586560,   6747586560,   5248122880,  3148873728,  1431306240,   477102080,   110100480,   15728640,   1048576;
  9: 4718592, 80216064, 641728512, 3208642560, 11230248960, 29198647296, 58397294592, 91767177216, 114708971520, 114708971520, 91767177216, 58397294592, 29198647296, 11230248960, 3208642560, 641728512, 80216064, 4718592;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (4*n+8*n*x+4*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 8 + 8*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 8 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 8*n*x + 4*n*x^2)*P[n - 1, x])/(n - 1)]];
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] = 8 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 8*n*x + 4*n*x^2)*v[i - 1])/(n - 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

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.