Real-rooted atlas

A196347 → x ↦ x+1 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
218483298
31626488643842058
41944103682073618432614457624
5291601944005184006912004608001228802016840
6524880419904013996800248832002488320013271040294912084707280
7110224801028764804115059209144576001219276800975421440433520640825753604150656720

All 15 rows

Technical details
Node id
n-a995131b65a4af0c · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-c91d5d506c107735

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:          3,           4;
  2:         18,          48,           32;
  3:        162,         648,          864,           384;
  4:       1944,       10368,        20736,         18432,          6144;
  5:      29160,      194400,       518400,        691200,        460800,        122880;
  6:     524880,     4199040,     13996800,      24883200,      24883200,      13271040,       2949120;
  7:   11022480,   102876480,    411505920,     914457600,    1219276800,     975421440,     433520640,      82575360;
  8:  264539520,  2821754880,  13168189440,   35115171840,   58525286400,   62426972160,   41617981440,   15854469120,   2642411520;
  9: 7142567040, 85710804480, 457124290560, 1422164459520, 2844328919040, 3792438558720, 3371056496640, 1926317998080, 642105999360, 95126814720;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+4*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[(3*n + 4*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] = (3*n + 4*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.