Real-rooted atlas

A013611 → D → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 4\,n x\right)\,P_{n-1}(x) + \left(4\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2403272
33006721921164
4200096646912102419600
512500118720160640563205120353300
675000134217630758401920000399360245766836952
7437500144428485291372852433920187443202580480114688141667484
825000001505550088521712641260607488689561600159416320155975685242883130933536
91406250015356194561314050227227943888896219636264967586611200122624409689653248235929673502567460
107812500015418694560196632230400586231244800636679680000308355317760732305817608737259520495452160104857601825869071720
11429687500152988826144287968315078411827706091520172867180851201128116183040036928189562886393770803205860884480026528972804613734447822191587500
122343750000150394711123241506730470656231916075809792447634804531200383101001564160166054251331584391316939735045155675176960374404546560138412032002013265921316394970795440
13126953125001467364058451259107022494374444509835615365121118866146492723212315402945331200687697540277862421185264819896323762273205616643897106104320022971783577607066563379287241523237973873515415604
14683593750001422761270114248338192677884160840333861725030402722093500103577603797658517616803842678241983435898881049130659411394562411717192633548833413937684807682792001372160001362450867814435420058419237580963841144978137692932088
Technical details
Node id
n-fa5dee57fd93f326 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-c7f57c7d3a795bc5

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:        0;
  1:        4;
  2:       40,         32;
  3:      300,        672,         192;
  4:     2000,       9664,        6912,        1024;
  5:    12500,     118720,      160640,       56320,        5120;
  6:    75000,    1342176,     3075840,     1920000,      399360,      24576;
  7:   437500,   14442848,    52913728,    52433920,    18744320,    2580480,     114688;
  8:  2500000,  150555008,   852171264,  1260607488,   689561600,  159416320,   15597568,   524288;
  9: 14062500, 1535619456, 13140502272, 27943888896, 21963626496, 7586611200, 1226244096, 89653248, 2359296;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (5*n+4*n*x)*P(n-1,x) + 4*n*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 4*n*x)*P[n - 1, x] + 4*n*x*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 4*n*x)*v[i - 1] + 4*n*x*deriv(v[i - 1], x))/(n - 1)); 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.