Real-rooted atlas

A075502 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-98 + 35\,x + 6\,x^{2}\right)\,P_{n-2}(x) + \left(-49\,x + 42\,x^{2}\right)\,P_{n-2}'(x) + \left(21 - 5\,x\right)\,P_{n-1}(x) + \left(7\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 70 + 23\,x + x^{2}\)
\(P_{1}(x) = 686 + 427\,x + 45\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
07023194
16864274511159
275467595143574116651
3912381361714042536051101271550
4117649025042431082851147490759515314918823
5157649664741254728487865552470142532014210203197629813
62157682669217799157476593951973477942122724110434062440226012104850680
729861669181831312569119720913625687128984598875581066908667227409039270324148949474241
84156421521036988692656352439677005123614154333044110727815393843473118215666745276006006039511220225362423
95802041614467561499233267140671809281858069162433101191462688494021899420158013217411361442946427839223088165473132425336681176
108110994299786155902327051835380492748026755275336776628914816675203153811171098475747289590137036438832093300984560577146796651251255581914157083909733
1111347087247379832334769443128111036628374968322594040996080244853445662545435110609446765941013578469779888003150762574960102572347878203396393224475451172627650127233542857394741
1215880108805707306733624280226248328412911305289485132189730549202317014441889323889163531114365677504109360512883387957324158313899720098129901469602483434253303953937988391946102325057491854340708443961894
13222280829895531261406125113783095587782570467784902450511048025489312680701603027219227769336015671638653994787102188637057034417991775445254328045388110685096434027658506946070559160856254730153223860642400306740855128137046911739497139
14311164676484684106294175154598663013552163741507584906245153802390892264498524342513142962253425299217818600723404557249813028916579860503686126331315460311962075376225369197753287404140090067071709963194870118388124800412903641478910802203974609681970212776488752849559
Technical details
Node id
n-6e894b02c6fc9b07 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8a154e29e4f61049

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:           70,            23,              1;
  1:          686,           427,             45,             1;
  2:         7546,          7595,           1435,            74,             1;
  3:        91238,        136171,          40425,          3605,           110,            1;
  4:      1176490,       2504243,        1082851,        147490,          7595,          153,           1;
  5:     15764966,      47412547,       28487865,       5524701,        425320,        14210,         203,         1;
  6:    215768266,     921779915,      747659395,     197347794,      21227241,      1043406,       24402,       260,       1;
  7:   2986166918,   18313125691,    19720913625,    6871289845,     988755810,     66908667,     2274090,     39270,     324,     1;
  8:  41564215210,  369886926563,   524396770051,  236141543330,   44110727815,   3938434731,   182156667,   4527600,   60060,   395,   1;
  9: 580204161446, 7561499233267, 14067180928185, 8069162433101, 1914626884940, 218994201580, 13217411361, 442946427, 8392230, 88165, 473, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(98-35*x-6*x^2)*P(n-2,x) - (49*x-42*x^2)*d/dx P(n-2,x) + (21-5*x)*P(n-1,x) + 7*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 70 + 23*x + x^2, P(1,x) = 686 + 427*x + 45*x^2 + x^3.
%t Mathematica
P[0, x_] := 70 + 23*x + x^2;
P[1, x_] := 686 + 427*x + 45*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(98 - 35*x - 6*x^2)*P[n - 2, x] - (49*x - 42*x^2)*D[P[n - 2, x], x] + (21 - 5*x)*P[n - 1, x] + 7*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] = 70 + 23*x + x^2; v[2] = 686 + 427*x + 45*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(98 - 35*x - 6*x^2)*v[i - 2] - (49*x - 42*x^2)*deriv(v[i - 2], x) + (21 - 5*x)*v[i - 1] + 7*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.