Real-rooted atlas

A038226 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + 16\,x + 8\,n x\right)\,P_{n}(x) = \left(9 + 96\,x + 192\,x^{2} + 24\,n x + 64\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 9 + 96\,x + 192\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0996192297
127432172820484235
281172810368245762048057233
3243648051840184320307200196608746691
47292332823328011059202764800353894418350089502009
5218781648979776580608019353600371589123853516816777216118694587
665612799363919104278691841161216002972712964624220164026531841509949441461537825
719683944784151165441254113286270566402006581248416179814454358179844076863488134217728017791787123
8590493149280566870405374771203135283200120394874883121348608054358179840611529523204026531840011811160064214573239881
917714710392624207852480221709312014780620800662171811842060090081284484549836806726824755206643777536003897682821121030792151042567805035499
105314413401222474826892888683724806651279360034054550323212360540487683228875882496605414227968079725330432007015829078016371085174374489335319756830528348755377
111594323110539728265295347234586652672288222105600166015932825668865868431362098769323622447222309781504777321971712009120577801420872361609003008348407747051527696581394432360919263322915
124782969357128352928533715213205812838412105328435207747410198528361545809264641259261594173443305561684705286529504562380809576606691491841013062526042112731656268808192323256418566144659706976665604246293593703513
131434890711479125603214155168049521798144049521798144003486334589337618077290463232070833464672256021250039401676804897128421785600861894602234265611396953417973760109748440321228807273269417738240296868139499520056294995342131249747228199404571
14430467213673320192110199605760182849716224019808719257600152130963898368867709942235136377778478252032012750023641006080335803091779584006895156817874124810941075281254809613169812838547456011637231068381184071248353479884800270215977642229764785074604081152580637495548775489
Technical details
Node id
n-e2b408f49345af0e · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9bf1c4f21fbefc3a

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      9,       96,       192;
  1:     27,      432,      1728,       2048;
  2:     81,     1728,     10368,      24576,       20480;
  3:    243,     6480,     51840,     184320,      307200,      196608;
  4:    729,    23328,    233280,    1105920,     2764800,     3538944,      1835008;
  5:   2187,    81648,    979776,    5806080,    19353600,    37158912,     38535168,     16777216;
  6:   6561,   279936,   3919104,   27869184,   116121600,   297271296,    462422016,    402653184,    150994944;
  7:  19683,   944784,  15116544,  125411328,   627056640,  2006581248,   4161798144,   5435817984,   4076863488,   1342177280;
  8:  59049,  3149280,  56687040,  537477120,  3135283200, 12039487488,  31213486080,  54358179840,  61152952320,  40265318400,  11811160064;
  9: 177147, 10392624, 207852480, 2217093120, 14780620800, 66217181184, 206009008128, 448454983680, 672682475520, 664377753600, 389768282112, 103079215104;
%F Formula
The row polynomials satisfy (3+(8*n+16)*x)*P(n,x) = (9+(24*n+96)*x+(64*n+192)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 96*x + 192*x^2.
%t Mathematica
P[0, x_] := 9 + 96*x + 192*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((9 + (24*n + 96)*x + (64*n + 192)*x^2)*P[n - 1, x])/(3 + (8*n + 16)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 9 + 96*x + 192*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((9 + (24*n + 96)*x + (64*n + 192)*x^2)*v[i - 1])/(3 + (8*n + 16)*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.