Real-rooted atlas

A225470 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 + 48\,x + 144\,x^{2} + 18\,n - 48\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-48\,x^{2} + 64\,x^{3} + 48\,n x^{2}\right)\,P_{n-2}'(x) + \left(-3 + 16\,x + 6\,n\right)\,P_{n-1}(x) + \left(-16\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131619
218144144306
31621584259210245362
419442160045360307206400106024
529160355104874800783360288000368642367288
65248806858432189423362032128010080000232243220070459250064
7110224801524251524595149445614479363386880001068318721685913610485761647838096

All 15 rows

Technical details
Node id
n-85048335272aedcf · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-86b851ff29f26c39

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:         18,          144,          144;
  3:        162,         1584,         2592,         1024;
  4:       1944,        21600,        45360,        30720,         6400;
  5:      29160,       355104,       874800,       783360,       288000,        36864;
  6:     524880,      6858432,     18942336,     20321280,     10080000,      2322432,      200704;
  7:   11022480,    152425152,    459514944,    561447936,    338688000,    106831872,    16859136,    1048576;
  8:  264539520,   3834563328,  12400185024,  16742412288,  11637561600,   4514807808,   986259456,  113246208,   5308416;
  9: 7142567040, 107765842176, 369316065600, 540224225280, 418854240000, 188932165632, 51209625600, 8210350080, 716636160, 26214400;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(48*n-48)*x-144*x^2)*P(n-2,x) + ((48*n-48)*x^2+64*x^3)*d/dx P(n-2,x) + (6*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[-(9*n^2 - 18*n + 9 + (48*n - 48)*x - 144*x^2)*P[n - 2, x] + ((48*n - 48)*x^2 + 64*x^3)*D[P[n - 2, x], x] + (6*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] = -(9*n^2 - 18*n + 9 + (48*n - 48)*x - 144*x^2)*v[i - 2] + ((48*n - 48)*x^2 + 64*x^3)*deriv(v[i - 2], x) + (6*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

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.