Real-rooted atlas

A038243 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(10 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110414
21009616212
310001744672643480
4100002841619072409625661840
51000004378244569601664002304010241185248
610000006529536997657654886401264640122880409624386368
7100000009541350420569651216065638454835200874496063078416384535993728
810000000013757890564084191232435722649620683407364816896005642649631457286553612526874880
9100000000019661046784790185984001121957478407120576512022724050944384525926434524364815335424262144310011309568
10100000000002792546549761500978958336278438084608023001328844809665445888002216350187522850029568020250624007340032010485768093526758400
11100000000000394956516966428134639869952672602944471047094097838080038196869201920114017689927681969551310848199053803520114766643203460300804194304222164548065280
12100000000000055693912375296522221778337792159226503731609621135066156892161429669989580800540447622561792120450025783296162384649912321324141772800632081612801610612736167772166392882423959552
1310000000000000783714773254144962276765958144037118717934305280613202321571840005134412615018086424093899125424128673889147001241611638156327649281258643815137288456975155200339805732864741552947267108864192330833547747328
14100000000000000110120068255580161763446769654824968551028651930419201742800907824005120178560471313416192010245690748651438083524534723621683207583582245617664010445024080691200926306283814912521774101954561789323640832338228674562684354566035248870815383552
Technical details
Node id
n-27f1ba1638e09efc · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-8a76f7e965c63a5f

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:         10,           4;
  2:        100,          96,          16;
  3:       1000,        1744,         672,           64;
  4:      10000,       28416,       19072,         4096,         256;
  5:     100000,      437824,      456960,       166400,       23040,        1024;
  6:    1000000,     6529536,     9976576,      5488640,     1264640,      122880,       4096;
  7:   10000000,    95413504,   205696512,    160656384,    54835200,     8744960,     630784,     16384;
  8:  100000000,  1375789056,  4084191232,   4357226496,  2068340736,   481689600,   56426496,   3145728,    65536;
  9: 1000000000, 19661046784, 79018598400, 112195747840, 71205765120, 22724050944, 3845259264, 345243648, 15335424, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (10+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[(10 + 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] = (10 + 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)

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.