Real-rooted atlas

A135065 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181624
236144144324
3128768153610243456
44003200960012800640032400
511521152046080921609216036864279936
63136376321881605017607526406021122007042286144
781921146886881282293760458752055050243670016104857617915904
82073633177623224329289728232243203715891237158912212336645308416136048896
951200921600737280034406400103219200206438400275251200235929600117964800262144001007769600
101239042478080223027201189478404163174409991618561665269760190316544014273740806343884801268776967316407296
1129491264880646488064038928384015571353604359979008871995801612457082880124570828808304721920332188876860397977652242776064
126922241661337618274713612183142405482414080175437250564093535846470174900224877186252807797211136046783266816170120970242835349504367876214784
131605632417464325009571843673686016183684300806612634828817633692876835267385753652901078630458778976256047023181004825649007820885496692736131533373442559896027136
143686400103219200134184960010734796800590413824002361655296007084965888001619420774400283398635520037786484736003778648473600274810798080013740539904004227858432006039797760017631936921600

Other recurrences

Image under the basis map A013611 · proved equal for all n (proof step)

\(P_{n}(x) = \left(4 + 16\,x + 16\,x^{2} + 16\,n + 64\,n x + 64\,n x^{2}\right)\,P_{n-2}(x) + \left(16 + 32\,x - 8\,n - 16\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 16\,x\)
Technical details
Node id
n-7af0d18f411a3a80 · revision 0 · JSON
Definition source
derived:basis_map_A013609
derived:basis_map_A013611 (equal by e-875231bfc71dcc22)
Fingerprint
f6-11aec07ce60b6d61

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:     8,     16;
  2:    36,    144,     144;
  3:   128,    768,    1536,     1024;
  4:   400,   3200,    9600,    12800,      6400;
  5:  1152,  11520,   46080,    92160,     92160,     36864;
  6:  3136,  37632,  188160,   501760,    752640,    602112,    200704;
  7:  8192, 114688,  688128,  2293760,   4587520,   5505024,   3670016,   1048576;
  8: 20736, 331776, 2322432,  9289728,  23224320,  37158912,  37158912,  21233664,   5308416;
  9: 51200, 921600, 7372800, 34406400, 103219200, 206438400, 275251200, 235929600, 117964800, 26214400;
%F Formula
The row polynomials satisfy n^2*P(n,x) = (2*n^2+4*n+2+(4*n^2+8*n+4)*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[Together[((2*n^2 + 4*n + 2 + (4*n^2 + 8*n + 4)*x)*P[n - 1, x])/(n^2)]];
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] = ((2*n^2 + 4*n + 2 + (4*n^2 + 8*n + 4)*x)*v[i - 1])/(n^2)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.