Real-rooted atlas

A038303 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 - 22\,x + 66\,n x + 4\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 726\,x + 132\,x^{2} + 8\,x^{3} + 726\,n x + 220\,n x^{2} + 16\,n x^{3} + 44\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111819
212117636333
31331290411881285551
4146414259226136563240089401
51610515856404791601548802200011521403883
6177156177304487906140340736072600076032313621620677
71948717199207416121754556655916801863400029272322414728192327851719
82143588811247178944178573348811544135684099480008586547210624768720896207364908864753
92357947691154338394322525537361619047823872811697040021251704323506173443568435220528645120072725531203
1025937424601188635815280347261387220299322946560148811124000467537495049641976960130842624011290752056320001239041067791513789
1128531167061122824933648884668736428180452725956672025723065720009428672816642333358424323957989376045539366403407360001499238429491215556800580191
1231384283767212738992037865661627320851976663998069785604244305843800017779783025664513338853350410449091952641502799091201499238400098949734438928384692224225122917189417
13345227121439313263965511789848011551710756889495172397934086743730396260003178136215837441048677943272962490366915379242980054008325359777280004716604006427833794569898803216056323238433832024859
143797498335832413866543760120272102814913621379961329324135710771210385344810240400543814419154406420187050408004485478807213834241103154719546881650811402240018158925424641428801454087622078464247267328368640046339891308614709
Technical details
Node id
n-5fb9021c553efde7 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c6eeb9717018ac53

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:         11,           8;
  2:        121,         176,          36;
  3:       1331,        2904,        1188,         128;
  4:      14641,       42592,       26136,        5632,        400;
  5:     161051,      585640,      479160,      154880,      22000,       1152;
  6:    1771561,     7730448,     7906140,     3407360,     726000,      76032,      3136;
  7:   19487171,    99207416,   121754556,    65591680,   18634000,    2927232,    241472,     8192;
  8:  214358881,  1247178944,  1785733488,  1154413568,  409948000,   85865472,  10624768,   720896,   20736;
  9: 2357947691, 15433839432, 25255373616, 19047823872, 8116970400, 2125170432, 350617344, 35684352, 2052864, 51200;
%F Formula
The row polynomials satisfy (121+(66*n-22)*x+4*n^2*x^2)*P(n,x) = (1331+(726*n+726)*x+(44*n^2+220*n+132)*x^2+(8*n^2+16*n+8)*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[((1331 + (726*n + 726)*x + (44*n^2 + 220*n + 132)*x^2 + (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x])/(121 + (66*n - 22)*x + 4*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] = ((1331 + (726*n + 726)*x + (44*n^2 + 220*n + 132)*x^2 + (8*n^2 + 16*n + 8)*x^3)*v[i - 1])/(121 + (66*n - 22)*x + 4*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.