Real-rooted atlas

A038226 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 8\,x + 8\,n\right)\,P_{n}(x) = \left(57 + 176\,x + 64\,x^{2} + 24\,n + 64\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 57 + 176\,x + 64\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
05717664297
1243136821125124235
294577762188822528409657233
334833780015552029184022528032768746691
4123931671849072002488320350208021626882621449502009
5430116940084681152169344003483648039223296201850882097152118694587
614652927527042220825699864576270950400445906944418381824184549376167772161461537825
749207510550088990973445329981441797562368390168576053508833284303355904166094438413421772817791787123
816336893936600042200352026425958401065996288028760997888520224768006115295232043033559040147639500801073741824214573239881
953734591437646321732104000123787699205813710848018761534668842182796902465399685120067268247552042077257728012992276070485899345922567805035499
101753755351585206469007023365542732800029709047808011162324828163001845547008578506928947278479622144007175279738880403941674188811338713661446871947673630528348755377
1156864187182390551226824307328239224347648144111052800061794819440641934802970214444598848126976752059007631369068756336640074622909284352381908491960329826885173248549755813888360919263322915
121833471456368788944102138708672100144080691266982817341443228087582720011535032962252830956847523430462438387377766493589565394124810157007097036807597968945315843564479258296328466239533875243980465111044246293593703513
13588305187220016574003821273366404085548346880300432242073601607587616194566456175165440001977434222100480464352712851456083251183170355201123074784729497611080371378585600759796894531584032902885461196807256776743321603518437208883249747228199404571
14187970681775303063936140810607360016304099696640130737547100160769106539708416342952024788172811805577445376000316389475536076806604127471665152010656151445805465613068506585943244811819062803824640074810771153879040300826381359513606192449487634432281474976710656580637495548775489
Technical details
Node id
n-faaea7a987614a48 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f7ea89bf2d0df11e

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:      57,       176,         64;
  1:     243,      1368,       2112,         512;
  2:     945,      7776,      21888,       22528,        4096;
  3:    3483,     37800,     155520,      291840,      225280,        32768;
  4:   12393,    167184,     907200,     2488320,     3502080,      2162688,       262144;
  5:   43011,    694008,    4681152,    16934400,    34836480,     39223296,     20185088,      2097152;
  6:  146529,   2752704,   22208256,    99864576,   270950400,    445906944,    418381824,    184549376,     16777216;
  7:  492075,  10550088,   99097344,   532998144,  1797562368,   3901685760,   5350883328,   4303355904,   1660944384,    134217728;
  8: 1633689,  39366000,  422003520,  2642595840, 10659962880,  28760997888,  52022476800,  61152952320,  43033559040,  14763950080,   1073741824;
  9: 5373459, 143764632, 1732104000, 12378769920, 58137108480, 187615346688, 421827969024, 653996851200, 672682475520, 420772577280, 129922760704, 8589934592;
%F Formula
The row polynomials satisfy (8*n+11+8*x)*P(n,x) = (24*n+57+(64*n+176)*x+64*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 57 + 176*x + 64*x^2.
%t Mathematica
P[0, x_] := 57 + 176*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((24*n + 57 + (64*n + 176)*x + 64*x^2)*P[n - 1, x])/(8*n + 11 + 8*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 57 + 176*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24*n + 57 + (64*n + 176)*x + 64*x^2)*v[i - 1])/(8*n + 11 + 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.