Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-7 + 4\,x + 8\,n\right)\,P_{n}(x) = \left(1 + 12\,x + 48\,x^{2} + 64\,x^{3} + 8\,n + 64\,n x + 128\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
19401665
21720886412802562625
3255044080166403456030720409690625
43392811200752643046407454721032192655360655362890625
5411480237602227201344000541900814622720255590402654208013107200104857687890625
64921604329652096041817602351923294617600272498688551485440749731840622854144251658240167772162587890625
757296871344104832010506496758517764059095041630666752492011520011021058048178436177921984744652813740539904469762048026843545674462890625
86539041094401899520228300802012774401344831488693501952027833794560869688934402099249152003847351173125190870630404885525299202899102924808589934592042949672962105712890625
9734968159120318566444651520465002496372586905623463198720117576695808471581327360151415842406438709063843847786305945600120747636817921396401242112011390253268992591416996659215461882265606871947673658746337890625
108161602219205034240806208009684418569049374720673146470404045229260801981336780800794295900569626065586094080697430546841601508660949811202601139568640003495931580252163537549813350402546056613068801175103052185602748779069440010995116277761621246337890625
1189748029937675891201367027201860655104198649651201704457666561194517463040691097042944033224816328704133145831669760444805704318976123510777184256028326409902489605310486543335424801150987141120095007081769205768545610529505280550305569701888022858846741463044837851162214401759218604441644345855712890625
12978928393024110105602203407363351355392402452807683912691875843132988784640209080140431361172219803729925548880984801282223042256175104753810246991872021585296596402176519451040043499521042411884411617281724346218777149442312341672689664002453516216917032961986179794647121921157213998001684484369899013432934484442493013196802814749767106561204013824462890625
13105105045044001547520034062080057254912007638478848082985943040074741235712005651666042880036203425955840019767576913510409235671180902400369925398986752001270309860671488003733155508912128009353628525107609601986136329578086400354283777708523520052436259284451328006328514051571712000607537348950884352044710540831883264002378463553205043200823314306878668800146366987889541120450359962737049632484531402587890625
1411312208635040211760645084352009359400960137338306560164894146560016499114311680139416384307200100452220796928062147774958796803317550305771520015327475022168064061381691540766720021307702981636915206402361697784299520166015621455151104003697434334630772736070246505972446003200112774120399007907840151016426739219824640165692004259332096000145232643733397176320979744025185615872004794982523261367091215321245932314427392252201579132747776072057594037927936871717929840087890625
Technical details
Node id
n-962e92ed2f9f6ef0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-803e02e9a42d02cb

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  9,   40,     16;
  2: 17,  208,    864,    1280,      256;
  3: 25,  504,   4080,   16640,    34560,     30720,       4096;
  4: 33,  928,  11200,   75264,   304640,    745472,    1032192,      655360,        65536;
  5: 41, 1480,  23760,  222720,  1344000,   5419008,   14622720,    25559040,     26542080,     13107200,       1048576;
  6: 49, 2160,  43296,  520960,  4181760,  23519232,   94617600,   272498688,    551485440,    749731840,     622854144,     251658240,      16777216;
  7: 57, 2968,  71344, 1048320, 10506496,  75851776,  405909504,  1630666752,   4920115200,  11021058048,   17843617792,   19847446528,   13740539904,     4697620480,      268435456;
  8: 65, 3904, 109440, 1899520, 22830080, 201277440, 1344831488,  6935019520,  27833794560,  86968893440,  209924915200,  384735117312,  519087063040,   488552529920,   289910292480,    85899345920,    4294967296;
  9: 73, 4968, 159120, 3185664, 44651520, 465002496, 3725869056, 23463198720, 117576695808, 471581327360, 1514158424064, 3870906384384, 7786305945600, 12074763681792, 13964012421120, 11390253268992, 5914169966592, 1546188226560, 68719476736;
%F Formula
The row polynomials satisfy (8*n-7+4*x)*P(n,x) = (8*n+1+(64*n+12)*x+(128*n+48)*x^2+64*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 1 + (64*n + 12)*x + (128*n + 48)*x^2 + 64*x^3)*P[n - 1, x])/(8*n - 7 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n + 1 + (64*n + 12)*x + (128*n + 48)*x^2 + 64*x^3)*v[i - 1])/(8*n - 7 + 4*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.