Real-rooted atlas

A028338 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 + 16\,x + 36\,x^{2} + 8\,n - 16\,n x - 4\,n^{2}\right)\,P_{n-2}(x) + \left(-16\,x^{2} + 16\,x^{3} + 16\,n x^{2}\right)\,P_{n-2}'(x) + \left(-2 + 8\,x + 4\,n\right)\,P_{n-1}(x) + \left(-8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
28483692
348352432128960
438432005040256040011584
53840350726480043520120001152160384
6460804515849354247526402800004838431362517248
76451206690816151280641386291262720001483776175616819244266496

All 15 rows

Technical details
Node id
n-2622d03bf3289660 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-12b23884bf72db83

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:         2,          8;
  2:         8,         48,         36;
  3:        48,        352,        432,        128;
  4:       384,       3200,       5040,       2560,        400;
  5:      3840,      35072,      64800,      43520,      12000,       1152;
  6:     46080,     451584,     935424,     752640,     280000,      48384,      3136;
  7:    645120,    6690816,   15128064,   13862912,    6272000,    1483776,    175616,     8192;
  8:  10321920,  112214016,  272157696,  275595264,  143673600,   41803776,   6849024,   589824,   20736;
  9: 185794560, 2102427648, 5403801600, 5928386560, 3447360000, 1166247936, 237081600, 28508160, 1866240, 51200;
%F Formula
The row polynomials satisfy P(n,x) = -(4*n^2-8*n+4+(16*n-16)*x-36*x^2)*P(n-2,x) + ((16*n-16)*x^2+16*x^3)*d/dx P(n-2,x) + (4*n-2+8*x)*P(n-1,x) - 8*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 8*n + 4 + (16*n - 16)*x - 36*x^2)*P[n - 2, x] + ((16*n - 16)*x^2 + 16*x^3)*D[P[n - 2, x], x] + (4*n - 2 + 8*x)*P[n - 1, x] - 8*x^2*D[P[n - 1, x], 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] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 - 8*n + 4 + (16*n - 16)*x - 36*x^2)*v[i - 2] + ((16*n - 16)*x^2 + 16*x^3)*deriv(v[i - 2], x) + (4*n - 2 + 8*x)*v[i - 1] - 8*x^2*deriv(v[i - 1], x)); 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.