Real-rooted atlas

A059364 → basis A013609 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(1 + 4\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1549
2457632153
3585152813283843825
499453533647024277766144126225
5208845940956169422415237766845441228805175225
6522112528536180649385447875577653684224195010562949120253586025
715141262597374072026822308164102196736376200422420686888966315540488257536014454403425
849966166253697864776011967331996822120387840025541643494418865086873687039328256229347164162642411520939536222625
9184874815125154808816562057591441581761249278301964817413747715328161750738012161001188641996839820003246089234190172169512681472068586144251625
107579867420125708666073954202980484371100167425698701326081213674977114368135972793446297610574902952673285637374701076481971401926901764083696009216038050725888005555477684381625
1134109403390562535225114992794001653031039534920046529775388808064872884482559813121145894560438661121074150924034007047189775914660659233675757355794432105118316825149441968054510551040167423193907200494437513909964625
121671360766137562518897557709216060097906576133752200030734138930283567366510563183205871360980472886243654246410763633417872207872867891463354695680051012045494710435842131516105521364992601002591777718272102670353007902720803631330754560047960438849266568625
138858212060529081251088481318427604430061717215359681017200213802355920054051008504877371147385732352858199915235842061312108031847199392722534410196914133073435197447216674020664070635523782329901651266109441426919748513300480003669366348186019430457647829617108582404178882919923712005035846079172989705625
1450491808745015763125670040339042697380100412743081382127646120015642898347585217870656407509420869240136005127719052795269659850649610963734815686100727808011862024499017850524467298237761062996439728128619726549551310123827202931449001577290294886410082289410140513763328238343778380168980070434664747849937321984023401744351572787200569050606946547836735625
Technical details
Node id
n-2fb0f21137a49f8a · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-9ac0ab5e73a51d73

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            1;
  1:            5,             4;
  2:           45,            76,            32;
  3:          585,          1528,          1328,            384;
  4:         9945,         35336,         47024,          27776,           6144;
  5:       208845,        940956,       1694224,        1523776,         684544,         122880;
  6:      5221125,      28536180,      64938544,       78755776,       53684224,       19501056,        2949120;
  7:    151412625,     973740720,    2682230816,     4102196736,     3762004224,     2068688896,      631554048,      82575360;
  8:   4996616625,   36978647760,  119673319968,   221203878400,   255416434944,   188650868736,    87039328256,   22934716416,   2642411520;
  9: 184874815125, 1548088165620, 5759144158176, 12492783019648, 17413747715328, 16175073801216, 10011886419968, 3982000324608, 923419017216, 95126814720;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+1+4*n*x)*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[(4*n + 1 + 4*n*x)*P[n - 1, 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] = (4*n + 1 + 4*n*x)*v[i - 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

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.