Real-rooted atlas

A013609 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9 + 6\,x\right)\,P_{n-1}(x) + \left(6\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19615
28114436261
3729264616202165211
465614406449896155521296117369
559049700326131220071928013608077762934711
6531441108591843175815627293760880632011197444665680415261
747829691660764067320763809274804564543711209650793688179842799362390393187
8430467212519843904163700624162943443059220559129696649003622497585689667184640167961676461270705
938742048938055938886358890374160892950000480855057863520376466190912828537776649281558016498845952100776962614452047775
103486784401573163606224776503349067626262992257920335746094990401981252862668859872171403529704821248008412356736036279705606046617695037325534197
1131381059609861837479976616650468494154075569099190789612655400670158409741362734350723882999430759688541789120691210617936088320733303549440259399895043627970563655617149866923
122824295364811294639083541443548308632570936214026848911624324629696816297009645584555066062848233183150790289926686112110008320111772970427571210990574014464061896805724161828497162242176782336148200700431185577
132541865828329194365320253104675291264734114520599162343856811280165621605872498776020555794565542716481322831508952681728480901608084598272103827265803765504136304398547665921086522400364544508518121512961273417666560130606940166310524802325233095
142287679245496129170049232935664159277847863159119616629130872538591680582501040010654637609010489515796652083271860894642196307648324629710660306022408803563799322223360148168130567288832015675268475375308810333020313534464408257703899136877678637875278364164096281382715067371309197
Technical details
Node id
n-5abce33cf81d1896 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-5162632e955115eb

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,           6;
  2:        81,         144,           36;
  3:       729,        2646,         1620,          216;
  4:      6561,       44064,        49896,        15552,         1296;
  5:     59049,      700326,      1312200,       719280,       136080,         7776;
  6:    531441,    10859184,     31758156,     27293760,      8806320,      1119744,       46656;
  7:   4782969,   166076406,    732076380,    927480456,    454371120,     96507936,     8817984,     279936;
  8:  43046721,  2519843904,  16370062416,  29434430592,  20559129696,   6490036224,   975856896,   67184640,   1679616;
  9: 387420489, 38055938886, 358890374160, 892950000480, 855057863520, 376466190912, 82853777664, 9281558016, 498845952, 10077696;
%F Formula
The row polynomials satisfy P(n,x) = (9+6*x)*P(n-1,x) + 6*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 + 6*x)*P[n - 1, x] + 6*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 + 6*x)*v[i - 1] + 6*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.