Real-rooted atlas

A225471 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 + 16\,x - 4\,x^{2} + 32\,n - 16\,n x - 16\,n^{2}\right)\,P_{n-2}(x) + \left(-4 + 4\,x + 8\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1628
25632492
373654412081408
4125441088032003841627024
52631682554889024014720112032624768
66561792693862427919365555205824030726416909248
7189628416214630400953774082189107228313602078728064128524574720

All 15 rows

Technical details
Node id
n-03c9a025d538894b · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7f75748c2b4416c3

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            1;
  1:            6,            2;
  2:           56,           32,            4;
  3:          736,          544,          120,           8;
  4:        12544,        10880,         3200,         384,         16;
  5:       263168,       255488,        90240,       14720,       1120,        32;
  6:      6561792,      6938624,      2791936,      555520,      58240,      3072,       64;
  7:    189628416,    214630400,     95377408,    21891072,    2831360,    207872,     8064,     128;
  8:   6233260032,   7461535744,   3588915200,   918994944,  138399744,  12644352,   688128,   20480,   256;
  9: 229682184192, 288263503872, 147994705920, 41364029440, 7004282880, 750575616, 51179520, 2150400, 50688, 512;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^2-32*n+16+(16*n-16)*x+4*x^2)*P(n-2,x) + (8*n-4+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^2 - 32*n + 16 + (16*n - 16)*x + 4*x^2)*P[n - 2, x] + (8*n - 4 + 4*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; v[2] = 6 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 32*n + 16 + (16*n - 16)*x + 4*x^2)*v[i - 2] + (8*n - 4 + 4*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.