Real-rooted atlas

A027467 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(256 - 48\,x + 144\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(4096 + 2304\,x + 432\,x^{2} + 27\,x^{3} + 2304\,n x + 720\,n x^{2} + 54\,n x^{3} + 144\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1161228
225638481721
340969216388843217632
465536196608124416276482025416233
5104857639321603317760110592016200087489575164
616777216754974727962624035389440777600083980835721215941897
7268435456140928614417836277769909043202903040004702924840007521399684793727664
842949672962576980377638050725888253671505929289728000200658124825604812817915904531441105053452273
968719476736463856467968782757789696608811614208267544166400722369249281229031014412899450887652750419683002277585190972
1010995116277768246337208320156551557939201391569403904071345111040002311581597696491612405760687970713606122200320314928000714492948929645121121
111759218604441614513553486643230614526885888030614526885888017938199347200067806393532416173047566827523027071139840359169085440277136640001257507504255091681042926639221728
122814749767106562533274790395904587798916209049665310990689894404305167843328000185983250831769655375221384806411623953176985617240116101120177367449600012072072038448977602568981352922077969594341401
13450359962737049643910096366862336111147431428620288135846860634980352994972123791360004835564521626009616454351497199616402963710135500871718882980659292231073792000836996994662450936706662418681214032312487308464563152486097372
147205759403792793675660473739824332820747520533342453762766336071112327168222873755729264640012035182809380290564607218419215892481289483872433602562677504964611276841319521058816004687183170109443803274097459220922959715846999715699210761680259723092439457645273
Technical details
Node id
n-6580f16d9b38761c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-4e5a1388374348f4

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:          16,           12;
  2:         256,          384,           81;
  3:        4096,         9216,         3888,          432;
  4:       65536,       196608,       124416,        27648,         2025;
  5:     1048576,      3932160,      3317760,      1105920,       162000,        8748;
  6:    16777216,     75497472,     79626240,     35389440,      7776000,      839808,       35721;
  7:   268435456,   1409286144,   1783627776,    990904320,    290304000,    47029248,     4000752,     139968;
  8:  4294967296,  25769803776,  38050725888,  25367150592,   9289728000,  2006581248,   256048128,   17915904,   531441;
  9: 68719476736, 463856467968, 782757789696, 608811614208, 267544166400, 72236924928, 12290310144, 1289945088, 76527504, 1968300;
%F Formula
The row polynomials satisfy (256+(144*n-48)*x+9*n^2*x^2)*P(n,x) = (4096+(2304*n+2304)*x+(144*n^2+720*n+432)*x^2+(27*n^2+54*n+27)*x^3)*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[((4096 + (2304*n + 2304)*x + (144*n^2 + 720*n + 432)*x^2 + (27*n^2 + 54*n + 27)*x^3)*P[n - 1, x])/(256 + (144*n - 48)*x + 9*n^2*x^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] = ((4096 + (2304*n + 2304)*x + (144*n^2 + 720*n + 432)*x^2 + (27*n^2 + 54*n + 27)*x^3)*v[i - 1])/(256 + (144*n - 48)*x + 9*n^2*x^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

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.