Real-rooted atlas

A088996 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1415
23220355
338440013615935
4614490884976119210521505
51228802370561813126862412828945623645
6294912070410246959104364140810627361637881039521827575
78257536023548723228638822419242777677094912184016322420304135135894930575

All 15 rows

Technical details
Node id
n-85bbd23e228e6595 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c294b27e66113b73

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:           4,            1;
  2:          32,           20,            3;
  3:         384,          400,          136,           15;
  4:        6144,         9088,         4976,         1192,          105;
  5:      122880,       237056,       181312,        68624,        12828,          945;
  6:     2949120,      7041024,      6959104,      3641408,      1062736,       163788,       10395;
  7:    82575360,    235487232,    286388224,    192427776,     77094912,     18401632,     2420304,     135135;
  8:  2642411520,   8774221824,  12696731648,  10453512192,   5353453824,   1745275904,   353474208,   40628880,   2027025;
  9: 95126814720, 360792981504, 606244110336, 592170876928, 370434044928, 153838647552, 42394761856, 7471701216, 763663860, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+(2*n-1)*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 + (2*n - 1)*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 + (2*n - 1)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.