Real-rooted atlas

A145324 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18311
2966812176
315361568532603696
4307204057620048439236096096
573728011888647651842457443938425202978976
620643840391864323093606413002304306870438563220160107243136
7660602880143976038413426319366944983042152192643995856041158081814404396968576

All 15 rows

Technical details
Node id
n-829809e4a0f73a58 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-e0b26d9c45ca69c1

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            1;
  1:            8,             3;
  2:           96,            68,            12;
  3:         1536,          1568,           532,            60;
  4:        30720,         40576,         20048,          4392,           360;
  5:       737280,       1188864,        765184,        245744,         39384,         2520;
  6:     20643840,      39186432,      30936064,      13002304,       3068704,       385632,       20160;
  7:    660602880,    1439760384,    1342631936,     694498304,     215219264,     39958560,     4115808,     181440;
  8:  23781703680,   58437402624,   62732353536,   38428258304,   14692876544,   3590700800,   547754688,   47689920,   1814400;
  9: 951268147200, 2599094845440, 3152105570304, 2227186221056, 1010425903104, 305249673984, 61407896320, 7932898368, 597165120, 19958400;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+4+(n+2)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4*n + 4 + (n + 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; for(n = 1, N, my(i = n + 1); v[i] = (4*n + 4 + (n + 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

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. [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.