Real-rooted atlas

A027465 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n x\right)\,P_{n}(x) = \left(4 + 8\,x + 4\,x^{2} + 4\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14812
2166448128
3643845762561280
4256204846084096128012288
51024102403072040960256006144114688
6409649152184320327680307200147456286721048576
71638422937610321922293760286720020643848028161310729437184
8655361048576550502414680064229376002202009612845056419430458982483886080
92621444718592283115528808038416515072019818086415414067275497472212336642621440738197504
101048576209715201415577605033164801101004800158544691215414067201006632960424673280104857600115343366442450944
1141943049227468869206016027682406406920601600116266106881356437913611072962560622854144023068672005075107845033164855834574848
121677721640265318433218887681476395008041523609600797253304321085150330881063004405767474249728036909875200121802588162415919104218103808481036337152
13671088641744830464157034741767677254041623991418880051821464780880611167436892127048499277732197171247982837760021112448614462813896704113413980169395240964123168604160
142684354567516192768732828794883908420239361343519457280322444669747256427817205767370163879936725500506931253740778291202955742806016117252607180831755914444852613349376402653184035184372088832
Technical details
Node id
n-3303313970cb18fb · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e99c3824c4864868

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:      4,       8;
  2:     16,      64,       48;
  3:     64,     384,      576,      256;
  4:    256,    2048,     4608,     4096,      1280;
  5:   1024,   10240,    30720,    40960,     25600,      6144;
  6:   4096,   49152,   184320,   327680,    307200,    147456,     28672;
  7:  16384,  229376,  1032192,  2293760,   2867200,   2064384,    802816,   131072;
  8:  65536, 1048576,  5505024, 14680064,  22937600,  22020096,  12845056,  4194304,   589824;
  9: 262144, 4718592, 28311552, 88080384, 165150720, 198180864, 154140672, 75497472, 21233664, 2621440;
%F Formula
The row polynomials satisfy (1+n*x)*P(n,x) = (4+(4*n+8)*x+(4*n+4)*x^2)*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[((4 + (4*n + 8)*x + (4*n + 4)*x^2)*P[n - 1, x])/(1 + n*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 + (4*n + 8)*x + (4*n + 4)*x^2)*v[i - 1])/(1 + n*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.