Real-rooted atlas

A085880 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-3\,n + 2\,n x + 2\,n^{2}\right)\,P_{n}(x) = \left(12 - 48\,x^{2} - 32\,n - 48\,n x + 32\,n x^{2} + 16\,n^{2} + 32\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
26410
3409632168
4280120014403203240
52016125442688021504358466528
614784120960376320537600322560430081415232
710982411151364561920946176010137600486604854067230792960
8823680999398450738688138378240215255040184504320738017287028736680524416

All 15 rows

Technical details
Node id
n-d06c736e578e5bae · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4be2d3b20c0d29e0

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        1;
   2:        6,         4;
   3:       40,        96,         32;
   4:      280,      1200,       1440,         320;
   5:     2016,     12544,      26880,       21504,        3584;
   6:    14784,    120960,     376320,      537600,      322560,       43008;
   7:   109824,   1115136,    4561920,     9461760,    10137600,     4866048,      540672;
   8:   823680,   9993984,   50738688,   138378240,   215255040,   184504320,    73801728,     7028736;
   9:  6223360,  87859200,  533012480,  1804042240,  3690086400,  4592107520,  3280076800,  1124597760,    93716480;
  10: 47297536, 761739264, 5376983040, 21746909184, 55203692544, 90333315072, 93678993408, 57354485760, 17206345728, 1274544128;
%F Formula
The row polynomials satisfy (2*n^2-3*n+2*n*x)*P(n,x) = (16*n^2-32*n+12+(32*n^2-48*n)*x+(32*n-48)*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 32*n + 12 + (32*n^2 - 48*n)*x + (32*n - 48)*x^2)*P[n - 1, x])/(2*n^2 - 3*n + 2*n*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((16*n^2 - 32*n + 12 + (32*n^2 - 48*n)*x + (32*n - 48)*x^2)*v[i - 1])/(2*n^2 - 3*n + 2*n*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.