Real-rooted atlas

A094638 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
12810
265696158
324368139215363320
4120260815792373763072086616
57202044817299264537611187207372802695536
6504017769619470721005337627192320373800962064384097399440
740320170380823039808153117696569406720121270272013862338566606028804006847808
83628801791475228914105623610910721124936832032779542528577503395845664197836823781703680184871442240
936288002052748803858807168374896814082187427814408137681367041954244182016294243375513625318171607049512681472009453831555456
10399168002548327680547633716486181895444484280654666240194511030128645947253238169612170364819865616025950774886412298915676160041855798476800530687942407296
1147900160034092610560825350086656106310590033928536828093952045943180253184017121603432407044451062549577728794744467881984093107791488614406454944768983040200907832688640032441760880542720
12622702080048918809088013184219086848191026172590080174765119241728010889826415030272479862654039121921517880376880005123436741585018552325449038451389235205755234210566635523641223988799078401044720729980928002145312383081591808
138717829120074962434355202227357383045123588472272949248368838179121971202615110042172620801332457184833273856497678772877878886413691038423774003200274853829898298982403922582783673971507237745587651533078528219748374922081075205850436087893196800152584569009152501760
1413076743680001222076201472003970720523151360704580192209756168044503277894041606401257626958577664370722433813602304001599290757610113597445189032032696737464321264167580104497561600228148700972461745766429625252971629133168642619521546576128376832141300827931002535936035102616527359180800011614920660773605840896
Technical details
Node id
n-57fc439b8bd33ef7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e22d44c4b2712e20

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:       2,         8;
  2:       6,        56,         96;
  3:      24,       368,       1392,        1536;
  4:     120,      2608,      15792,       37376,        30720;
  5:     720,     20448,     172992,      645376,      1118720,       737280;
  6:    5040,    177696,    1947072,    10053376,     27192320,     37380096,      20643840;
  7:   40320,   1703808,   23039808,   153117696,    569406720,   1212702720,    1386233856,     660602880;
  8:  362880,  17914752,  289141056,  2361091072,  11249368320,  32779542528,   57750339584,   56641978368,   23781703680;
  9: 3628800, 205274880, 3858807168, 37489681408, 218742781440, 813768136704, 1954244182016, 2942433755136, 2531817160704, 951268147200;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(8*n^3-8*n^2)*x+(16*n^3-32*n^2+16*n)*x^2)*P(n-2,x) + (2*n^2-1+(8*n^2-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (8*n^3 - 8*n^2)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (8*n^2 - 8*n)*x)*P[n - 1, x])/(n - 1)]];
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] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (8*n^3 - 8*n^2)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (8*n^2 - 8*n)*x)*v[i - 1])/(n - 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.