Real-rooted atlas

A136124 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 16\,x + 144\,x^{2} - 2\,n - 16\,n x - n^{2}\right)\,P_{n-2}(x) + \left(16\,x^{2} + 64\,x^{3} + 16\,n x^{2}\right)\,P_{n-2}'(x) + \left(3 + 16\,x + 2\,n\right)\,P_{n-1}(x) + \left(-16\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131619
212112144268
360752172810243564
436054721713618432640047800
525204406416920025088016000036864663528
62016039283217501763210240284800012165122007049638624
718144038580481928707241337856456960002735308884295681048576147191648

All 15 rows

Technical details
Node id
n-33649b0b52c81020 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-dcd95a62876f0c72

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:        3,        16;
  2:       12,       112,        144;
  3:       60,       752,       1728,       1024;
  4:      360,      5472,      17136,      18432,        6400;
  5:     2520,     44064,     169200,     250880,      160000,       36864;
  6:    20160,    392832,    1750176,    3210240,     2848000,     1216512,     200704;
  7:   181440,   3858048,   19287072,   41337856,    45696000,    27353088,    8429568,    1048576;
  8:  1814400,  41483520,  227593152,  550531072,   715321600,   536739840,  233218048,   54525952,   5308416;
  9: 19958400, 485349120, 2876876352, 7674281984, 11309356800, 10024390656, 5487648768, 1818230784, 334430208, 26214400;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+2*n+1+(16*n+16)*x-144*x^2)*P(n-2,x) + ((16*n+16)*x^2+64*x^3)*d/dx P(n-2,x) + (2*n+3+16*x)*P(n-1,x) - 16*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 16*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 2*n + 1 + (16*n + 16)*x - 144*x^2)*P[n - 2, x] + ((16*n + 16)*x^2 + 64*x^3)*D[P[n - 2, x], x] + (2*n + 3 + 16*x)*P[n - 1, x] - 16*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] = 3 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 2*n + 1 + (16*n + 16)*x - 144*x^2)*v[i - 2] + ((16*n + 16)*x^2 + 64*x^3)*deriv(v[i - 2], x) + (2*n + 3 + 16*x)*v[i - 1] - 16*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.