Real-rooted atlas

A136572 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1415
23216250
3384288726750
46144614423043842415000
512288015360076800192002400120375000
62949120442368027648009216001728001728072011250000
782575360144506880108380160451584001128960016934401411205040393750000

All 15 rows

Technical details
Node id
n-88c0cd8da579d1de · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-03d7b8417365c509

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,           16,            2;
  3:         384,          288,           72,            6;
  4:        6144,         6144,         2304,          384,          24;
  5:      122880,       153600,        76800,        19200,        2400,         120;
  6:     2949120,      4423680,      2764800,       921600,      172800,       17280,        720;
  7:    82575360,    144506880,    108380160,     45158400,    11289600,     1693440,     141120,      5040;
  8:  2642411520,   5284823040,   4624220160,   2312110080,   722534400,   144506880,   18063360,   1290240,    40320;
  9: 95126814720, 214035333120, 214035333120, 124853944320, 46820229120, 11705057280, 1950842880, 209018880, 13063680, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+n*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 + n*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 + n*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.