Real-rooted atlas

A204420 → basis A007318 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-3\,x - 2\,n + 6\,n x + 4\,n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1235
2245427105
3720198016204054725
440320126000132300567008505363825
5362880012428640153090008675100229635022963542567525
647900160017603308802430933120165031020058939650010609137075779557023641625
7871782912003390612825605110827321603951628480801716322608004229509284055167512402955402451552224799125

All 15 rows

Technical details
Node id
n-81c130d5be6f4626 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-570085136c45352b

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:                2,                 3;
  2:               24,                54,                27;
  3:              720,              1980,              1620,               405;
  4:            40320,            126000,            132300,             56700,              8505;
  5:          3628800,          12428640,          15309000,           8675100,           2296350,           229635;
  6:        479001600,        1760330880,        2430933120,        1650310200,         589396500,        106091370,          7577955;
  7:      87178291200,      339061282560,      511082732160,      395162848080,      171632260800,      42295092840,       5516751240,       295540245;
  8:   20922789888000,    85297730918400,   137917613433600,   117837806486400,    58974070755600,   17874274017600,    3227299475400,    319183464600,    13299311025;
  9: 6402373705728000, 27168167945318400, 46552973987520000, 43092167069952000, 24055793782020000, 8477205457921200, 1899141614370000, 262262413413000, 20347945868250, 678264862275;
%F Formula
The row polynomials satisfy P(n,x) = (4*n^2-2*n+(6*n-3)*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 - 2*n + (6*n - 3)*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 - 2*n + (6*n - 3)*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.