Real-rooted atlas

A038226 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(19 + 8\,x + 8\,n\right)\,P_{n}(x) = \left(297 + 304\,x + 64\,x^{2} + 88\,n + 64\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 297 + 304\,x + 64\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
029730464665
142357128364851215523
257233135520114048389124096349809
3746691228932027104001520640389120327687688939
49502009358411685494368043366400182476803735552262144165898633
5118694587532112504100355270410256153606071296002043740163486515220971523528441075
614615378257596453568170276001282140912435216409845760777125888021799895043187671041677721674191354337
717791787123105230723400273472328448408662403072385364238336236301778944932551065602242274918428689039361342177281545504236731
8214573239881142334296984042092289360007292595425280817324806144061658278133763150690385920106577264640022422749184025501368320107374182431946082080121
92567805035499188824451095286262709067296012347071545600016043709935616014384916588134490432141262848396086791372801172349911040021924465868802244120412168589934592656024089584707
1030528348755377246509283407904906357365257344200406690153472029632971709440003080392307638272230158665410150412402122230333444753041496473601250506571776002104748723404819585050869766871947673613396379773295185
113609192633229153174948270559208128184827372110083142038866225459252105739439902720616365811556352005339346666573004834195001718079488161227588994334725492403507036160130052683464704019899442475827216973710753792549755813888272237735045137803
12424629359370351340422957492166480177797103151315648478556688855877632879770882543128576116716856345382092811505495149051904008542954666516807684787300240531128322006387774151720966151491927880499213241727770951680185728129774387214623504649420843980465111045508940833555674729
13497472281994045715095552312444215602425377449529988800711188412605262592014356700665676328960211145011810350858242334337126907641856019723705969803264000128144319997752115206383066987374837760240766532898206515267107184667787264013241727770951680017144135056097280125344325566464035184372088832111061929323820892819
14580637495548775489636764520952378508832611534799642979840103482771179946188800227580292033684029440367531537041314021376450442691862081830912426850217491683082240315579295516852224000182249699552358563840817032574383979233282801646928270039449671580996978973081601303800888216780800156746377655746560106960491150049282814749767106562231625673391595349953
Technical details
Node id
n-8f7968e2e6e48a72 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3845cffbc8d69491

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:           297,            304,             64;
  1:          4235,           7128,           3648,             512;
  2:         57233,         135520,         114048,           38912,            4096;
  3:        746691,        2289320,        2710400,         1520640,          389120,           32768;
  4:       9502009,       35841168,       54943680,        43366400,        18247680,         3735552,         262144;
  5:     118694587,      532112504,     1003552704,      1025615360,       607129600,       204374016,       34865152,        2097152;
  6:    1461537825,     7596453568,    17027600128,     21409124352,     16409845760,      7771258880,     2179989504,      318767104,       16777216;
  7:   17791787123,   105230723400,   273472328448,    408662403072,    385364238336,    236301778944,    93255106560,    22422749184,     2868903936,     134217728;
  8:  214573239881,  1423342969840,  4209228936000,   7292595425280,   8173248061440,   6165827813376,  3150690385920,  1065772646400,   224227491840,   25501368320,   1073741824;
  9: 2567805035499, 18882445109528, 62627090672960, 123470715456000, 160437099356160, 143849165881344, 90432141262848, 39608679137280, 11723499110400, 2192446586880, 224412041216, 8589934592;
%F Formula
The row polynomials satisfy (8*n+19+8*x)*P(n,x) = (88*n+297+(64*n+304)*x+64*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 297 + 304*x + 64*x^2.
%t Mathematica
P[0, x_] := 297 + 304*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((88*n + 297 + (64*n + 304)*x + 64*x^2)*P[n - 1, x])/(8*n + 19 + 8*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 297 + 304*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((88*n + 297 + (64*n + 304)*x + 64*x^2)*v[i - 1])/(8*n + 19 + 8*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.