Real-rooted atlas

A196347 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
15914
250180162392
375040507290437416464
415000108000291600349920157464921984
53750003375000121500002187000019683000708588064538880
6112500001215000005467500001312200000177147000012754584003826375205421265920
7393750000496125000026790750000803722500001446700500001562436540009374619240024106163760531284060160

All 15 rows

Technical details
Node id
n-41ebfdeeaf308611 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-dca555f310728b4d

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:            5,              9;
  2:           50,            180,            162;
  3:          750,           4050,           7290,            4374;
  4:        15000,         108000,         291600,          349920,          157464;
  5:       375000,        3375000,       12150000,        21870000,        19683000,          7085880;
  6:     11250000,      121500000,      546750000,      1312200000,      1771470000,       1275458400,        382637520;
  7:    393750000,     4961250000,    26790750000,     80372250000,    144670050000,     156243654000,      93746192400,      24106163760;
  8:  15750000000,   226800000000,  1428840000000,   5143824000000,  11573604000000,   16665989760000,   14999390784000,    7713972403200,   1735643790720;
  9: 708750000000, 11481750000000, 82668600000000, 347208120000000, 937461924000000, 1687431463200000, 2024917755840000, 1562079411648000, 702935735241600, 140587147048320;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+9*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[(5*n + 9*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] = (5*n + 9*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.