Real-rooted atlas

A075503 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-128 + 48\,x + 7\,x^{2}\right)\,P_{n-2}(x) + \left(-64\,x + 56\,x^{2}\right)\,P_{n-2}'(x) + \left(24 - 6\,x\right)\,P_{n-1}(x) + \left(8\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 88 + 26\,x + x^{2}\)
\(P_{1}(x) = 960 + 544\,x + 51\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
088261115
19605445111556
21177610880184084124581
31597442191365856046401251442206
4232652845301761770496215040980017418852215
53538944096600064525557769117696622720183682311194304296
6551550976212074496015568076803684925443524505615321603158429614634405257
7870737510447706013696463876915201451425792018609561601115627523346560508803691119291254962
81383784775681093639602176139514714521656437506048094085939200745397452830466867266739207788045013293391620091
9220653944832025424595779584423911762165762183053023641646277486182404703385026562511813427274261299212386880114400539196976802050876
103524450163097659701119156224013006489727795208437973951447042237127836303362853779275776019339895111687402664755216537297922169024016244863613030962329247373
1156343098975846414117282624045056402513543875788803267317181972480010700208807608320168499172134092814182527270912068281827328001960173895683421530112361961602242247411100139293280839174
1290110475454382083354380453292277761254720218161217536126907186856067072050863173355438080097609745078157312100376521872506885950801549721602133169831936047552913408066595809285800704030212085413485137204516659583
131441459744014336007996477588954415104393439751410821365764947075108001388953624099230279333314560557952599092258406469201669363911884849703312789864448217843470434304060273754112000107242930995212310724608898060803987209751127378065854425452112
1423060892998183157761910508325791019827201239520398307345039360193553250427785864806411403203686139334492163159964505547472896004678518659712496435240191253475418439682119289238171156487126983109509120156589535657984227369258188821775423488134955520516800110414875750170369537294609
Technical details
Node id
n-d741cd559e72192f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-c9f198fd808e3a9a

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:            88,             26,              1;
  1:           960,            544,             51,              1;
  2:         11776,          10880,           1840,             84,             1;
  3:        159744,         219136,          58560,           4640,           125,            1;
  4:       2326528,        4530176,        1770496,         215040,          9800,          174,           1;
  5:      35389440,       96600064,       52555776,        9117696,        622720,        18368,         231,         1;
  6:     551550976,     2120744960,     1556807680,      368492544,      35245056,      1532160,       31584,       296,        1;
  7:    8707375104,    47706013696,    46387691520,    14514257920,    1860956160,    111562752,     3346560,     50880,      369,      1;
  8:  138378477568,  1093639602176,  1395147145216,   564375060480,   94085939200,   7453974528,   304668672,   6673920,    77880,    450,   1;
  9: 2206539448320, 25424595779584, 42391176216576, 21830530236416, 4627748618240, 470338502656, 25118134272, 742612992, 12386880, 114400, 539, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(128-48*x-7*x^2)*P(n-2,x) - (64*x-56*x^2)*d/dx P(n-2,x) + (24-6*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 88 + 26*x + x^2, P(1,x) = 960 + 544*x + 51*x^2 + x^3.
%t Mathematica
P[0, x_] := 88 + 26*x + x^2;
P[1, x_] := 960 + 544*x + 51*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(128 - 48*x - 7*x^2)*P[n - 2, x] - (64*x - 56*x^2)*D[P[n - 2, x], x] + (24 - 6*x)*P[n - 1, x] + 8*x*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] = 88 + 26*x + x^2; v[2] = 960 + 544*x + 51*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(128 - 48*x - 7*x^2)*v[i - 2] - (64*x - 56*x^2)*deriv(v[i - 2], x) + (24 - 6*x)*v[i - 1] + 8*x*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.