Real-rooted atlas

A038221 → x ↦ x+1 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9 + 3\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19312
281639153
3729999324272079
4656114175785714588130132
559049189783160380498156075243465345
6531441245454329750491377810277020240577297640649
7478296931048839519893643372572799508501408428918542187133000218
84304672138693497587298697776303117831014503163654822670534234117265612446852779
938742048947723598631425560958016353522135880213918524581508213720087003035118612400291968347432821671
1034867844015843057982322815122328933713022717023390548929085402037259174186873632026561680131974515442867559049966088052514
11313810596097116273110793597560088804675279775892759234059566002751365362086726510670578113052912489104348440355553558451558893617714720618707086135
122824295364818633670911775560982832652971323430399270981446499183653818380298655986427869884191864557111938640468350859062251297342525227403603954206982531441459988466851773
13254186582832910445133955074386737526171478025504695684836553434677405454295244522143253287910038958328599202507432341677123926914164973838972301027662425794330969098801361186535791159432310702794178709388
142287679245496112610416720939031332398294437392948510578017850130797796342199911606898934659715198134440851784816477105339845236111202048111376944490258107708648790211522881990421109887956450356905146786361348774782969259190814145524957
Technical details
Node id
n-7f135bf281132ad6 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-322b0642acfc2857

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:         9,          3;
  2:        81,         63,           9;
  3:       729,        999,         324,          27;
  4:      6561,      14175,        7857,        1458,         81;
  5:     59049,     189783,      160380,       49815,       6075,        243;
  6:    531441,    2454543,     2975049,     1377810,     277020,      24057,       729;
  7:   4782969,   31048839,    51989364,    33725727,    9950850,    1408428,     91854,     2187;
  8:  43046721,  386934975,   872986977,   763031178,  310145031,   63654822,   6705342,   341172,    6561;
  9: 387420489, 4772359863, 14255609580, 16353522135, 8802139185, 2458150821, 372008700, 30351186, 1240029, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (9+3*x)*P(n-1,x) + 3*x*d/dx 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[(9 + 3*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], 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] = (9 + 3*x)*v[i - 1] + 3*x*deriv(v[i - 1], 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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.