Real-rooted atlas

A013611 → basis A008277

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1549
225561697
3125604432641225
462559368032281625617665
531255592412816080000166401024284873
615625515816188977618726406694409216040965059553
778125470484426630352393939842154880049817604874241638497841673
839062542656096365013952776219136610100736210739200340623362490368655362041737985
9195312538546736449158057601465578112015916992000770888294418307645442184314881238630426214445646726793
109765625347701877665447344336268811502080392879956480256390041600839277035521453129728013320192006029312010485761086867990625
11488281253133223148486472355147248315849127049325725094400798127086592034594635694088152436244481074098995207800094720288358400419430427424895224905
12244140625282185395856113667350950728559583772185621516656663321623683467202560013224952697651240740893884416723514078003274944348160044176506880135895449616777216730266798373377
13122070312525406451252041488962978193601500596181651840486088125018496067815330671728644782574970421248187344760609177643066378439884859667745865728498571673600024329479782463250104326710886420446038173088073
146103515625228706889393361945814452152496261057202793587201080808909804915201889818516800614401658208064109076488095407088271360023428350447124480416903271809024046302823658291231864312037376130840474419229125246976268435456600005644990402017

Computed in your browser ?

Technical details
Node id
n-e6ac71558d9546a6 · revision 1 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-c7f57c7d3a795bc5

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       5,         4;
  2:      25,        56,         16;
  3:     125,       604,        432,          64;
  4:     625,      5936,       8032,        2816,         256;
  5:    3125,     55924,     128160,       80000,       16640,       1024;
  6:   15625,    515816,    1889776,     1872640,      669440,      92160,       4096;
  7:   78125,   4704844,   26630352,    39393984,    21548800,    4981760,     487424,     16384;
  8:  390625,  42656096,  365013952,   776219136,   610100736,  210739200,   34062336,   2490368,    65536;
  9: 1953125, 385467364, 4915805760, 14655781120, 15916992000, 7708882944, 1830764544, 218431488, 12386304, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (5+4*x)*P(n-1,x) + 4*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[(5 + 4*x)*P[n - 1, x] + 4*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] = (5 + 4*x)*v[i - 1] + 4*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)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-dd742906c5111887
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-dd742906c5111887
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-dd742906c5111887

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

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. [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
  4. [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
  5. [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