Real-rooted atlas

A094638 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
21821645
316226113824585
41944394229619781209945
5291607079468067324367668720208845
65248801478412172076410603173650766663650405221125
71102248035245692479633403603276916149132431996463892840320151412625

All 15 rows

Technical details
Node id
n-b93b6a8b17cd0e40 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-39130ef25fb55f32

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,           2;
  2:         18,          21,           6;
  3:        162,         261,         138,          24;
  4:       1944,        3942,        2961,         978,         120;
  5:      29160,       70794,       68067,       32436,        7668,         720;
  6:     524880,     1478412,     1720764,     1060317,      365076,       66636,       5040;
  7:   11022480,    35245692,    47963340,    36032769,    16149132,     4319964,     638928,     40320;
  8:  264539520,   945098928,  1468331388,  1296456516,   711874089,   249021324,   54213948,   6718032,   362880;
  9: 7142567040, 28163066256, 49095936756, 49687639812, 32185165563, 13842316638, 3953989836, 723526344, 76978080, 3628800;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+(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[(3*n + (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] = (3*n + (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. [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.