Real-rooted atlas

A225471 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 + 64\,x + 144\,x^{2} + 32\,n - 64\,n x - 16\,n^{2}\right)\,P_{n-2}(x) + \left(-64\,x^{2} + 64\,x^{3} + 64\,n x^{2}\right)\,P_{n-2}'(x) + \left(-4 + 16\,x + 8\,n\right)\,P_{n-1}(x) + \left(-16\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141620
232192144368
33842816345610247680
461445120080640409606400185344
5122880112230420736001392640384000368645132288
62949120289013765986713648168960179200003096576200704161103872
782575360856424448193639219217744527368028160001899233282247884810485765666111488
82642411520287267880966967237017670552387584367804416001070176665617533501441509949445308416220985819136
995126814720107644295577627667464192003035333918720176504832000059711894323212138577920014596177920955514880262144009472781058048
103805072588800445797472665601203578433699841410879979520008957276979200034051436052480810641208115212180258816001121137459205767168000126877696442897312776192
111674231939072002022390041149440569696283367833670637499071856644823001858048000201420234031104054207328341196895945005989888110992608460808074035200003349571174460397977622437689227739136
12803631330754560099753493077688320291655726386905088379571731251068928275652526106214400124462203037286400369857303454679047437416625340416101848611343564893566533632000551562720051218844169011228353495041224672899688824832
134178882919923712005315762652960522240160638792098182594562181172630158468710416706254677842329600805979310831068774426008855278340014085799772353021870089061319956443955298950208028672007396746372055043605517670809610320672194561315333734471657452001046822912
14234017443515727872003043689212376671846409474190996264672296961335688702825227878400107187355134407475200054757644100977033216018953068537061782323246067025077442969600801047392932593664010015937554022400008931368036938547255401856892928002270547882803205524401684480603979776004475033325033215754240
Technical details
Node id
n-52e9f202925df115 · 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:           4,            16;
  2:          32,           192,           144;
  3:         384,          2816,          3456,          1024;
  4:        6144,         51200,         80640,         40960,          6400;
  5:      122880,       1122304,       2073600,       1392640,        384000,        36864;
  6:     2949120,      28901376,      59867136,      48168960,      17920000,      3096576,       200704;
  7:    82575360,     856424448,    1936392192,    1774452736,     802816000,    189923328,     22478848,     1048576;
  8:  2642411520,   28726788096,   69672370176,   70552387584,   36780441600,  10701766656,   1753350144,   150994944,   5308416;
  9: 95126814720, 1076442955776, 2766746419200, 3035333918720, 1765048320000, 597118943232, 121385779200, 14596177920, 955514880, 26214400;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^2-32*n+16+(64*n-64)*x-144*x^2)*P(n-2,x) + ((64*n-64)*x^2+64*x^3)*d/dx P(n-2,x) + (8*n-4+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) = 4 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 16*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^2 - 32*n + 16 + (64*n - 64)*x - 144*x^2)*P[n - 2, x] + ((64*n - 64)*x^2 + 64*x^3)*D[P[n - 2, x], x] + (8*n - 4 + 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] = 4 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 32*n + 16 + (64*n - 64)*x - 144*x^2)*v[i - 2] + ((64*n - 64)*x^2 + 64*x^3)*deriv(v[i - 2], x) + (8*n - 4 + 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.