Real-rooted atlas

A137320 → basis A013609 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2203252
3104288192584
46162272268810246600
54176185602976020480512078096
63211216332831968030208013824024576980016
7277056156352035374084121600261632086016011468813090752
8265478416288128412177925596057644154880202997765046272524288186146496
9279993601841195525099066887794339847213516804149411841451950082831155223592962813618304
10322508160224930995267136908801129327616011832038400802684108835335372809751756801533542401048576045110217600
11402976512029571784704940601502721713256652801984508416001531158773767986103910427855421440622854144080740352046137344765352627200
12543029529604166713221121400212661760272888778342434364892672002943070445568175273426944072719951462420632043520038178652160415236096020132659213708420992000
1378514942464062678221332482210300519731245672229830656617552886579205769813472051238272608518144181932927221766160883908608145148084224022611912294420937965568872415232258607825459200
141212595453440010029852739584036915201075916880296614184550411546904790220801161897406365696843347089801216447859886915584174283075878912491539188940809778302550016130077111091210381741260837580963845126961140582400
Technical details
Node id
n-7fd1278b59a5838a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a7b8cffa591f9e51

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:        0;
  1:        4;
  2:       20,        32;
  3:      104,       288,       192;
  4:      616,      2272,      2688,      1024;
  5:     4176,     18560,     29760,     20480,      5120;
  6:    32112,    163328,    319680,    302080,    138240,     24576;
  7:   277056,   1563520,   3537408,   4121600,   2616320,    860160,    114688;
  8:  2654784,  16288128,  41217792,  55960576,  44154880,  20299776,   5046272,   524288;
  9: 27999360, 184119552, 509906688, 779433984, 721351680, 414941184, 145195008, 28311552, 2359296;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+8*n*x+16*x^2)*P(n-2,x) + (2*n+1+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 8*n*x + 16*x^2)*P[n - 2, x] + (2*n + 1 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 8*n*x + 16*x^2)*v[i - 2] + (2*n + 1 + 8*x)*v[i - 1]); 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.