Real-rooted atlas

A137320 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
26403682
324208324128684
41201232255617924006100
5720835220880198408000115258944
6504064224183744213120118000311043136618368
74032055411217589602358272161000058867210976081927028288

All 15 rows

Technical details
Node id
n-73e25ae24eadc15f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c68171648d5b88d8

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:       6,       40,        36;
  3:      24,      208,       324,       128;
  4:     120,     1232,      2556,      1792,       400;
  5:     720,     8352,     20880,     19840,      8000,      1152;
  6:    5040,    64224,    183744,    213120,    118000,     31104,     3136;
  7:   40320,   554112,   1758960,   2358272,   1610000,    588672,   109760,     8192;
  8:  362880,  5309568,  18324144,  27478528,  21859600,   9934848,  2590336,   360448,   20736;
  9: 3628800, 55998720, 207134496, 339937792, 304466400, 162304128, 52948224, 10371072, 1119744, 51200;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+8*n*x-36*x^2)*P(n-2,x) + (8*n*x^2+16*x^3)*d/dx P(n-2,x) + (2*n+1+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[-(n^2 + 8*n*x - 36*x^2)*P[n - 2, x] + (8*n*x^2 + 16*x^3)*D[P[n - 2, x], x] + (2*n + 1 + 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] = -(n^2 + 8*n*x - 36*x^2)*v[i - 2] + (8*n*x^2 + 16*x^3)*deriv(v[i - 2], x) + (2*n + 1 + 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

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.