Real-rooted atlas

A013622 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(20 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
120828
240032064784
380009600384051221952
4160000256000153600409604096614656
532000006400000512000020480004096003276817210368
66400000015360000015360000081920000245760003932160262144481890304
71280000000358400000043008000002867200000114688000027525120036700160209715213492928512
82560000000081920000000114688000000917504000004587520000014680064000293601280033554432016777216377801998336
9512000000000184320000000029491200000002752512000000165150720000066060288000017616076800030198988800301989888013421772810578455953408
1010240000000000409600000000007372800000000078643200000000550502400000002642411520000088080384000002013265920000301989888000268435456001073741824296196766695424
1120480000000000090112000000000018022400000000002162688000000000173015040000000096888422400000038755368960000011072962560000022145925120000295279001600023622320128085899345928293509467471872
1240960000000000001966080000000000043253760000000000576716800000000005190451200000000033218887680000000155021475840000005315022028800000132875550720000023622320128000028346784153600206158430208068719476736232218265089212416
1381920000000000000425984000000000000102236160000000000014994636800000000001499463680000000000107961384960000000057579405312000000023031762124800000069095286374400000153545080832000002456721293312000268005959270400178670639513605497558138886502111422497947648
14163840000000000000091750400000000000002385510400000000000038168166400000000000419849830400000000003358798643200000000020152791859200000000921270484992000000032244466974720000008598524526592000001719704905318400002501388953190400025013889531904001539316278886404398046511104182059119829942534144
Technical details
Node id
n-94c47e4fba64d6ad · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-9629d1e9c04dbfbe

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            1;
  1:           20,             8;
  2:          400,           320,            64;
  3:         8000,          9600,          3840,           512;
  4:       160000,        256000,        153600,         40960,          4096;
  5:      3200000,       6400000,       5120000,       2048000,        409600,        32768;
  6:     64000000,     153600000,     153600000,      81920000,      24576000,      3932160,       262144;
  7:   1280000000,    3584000000,    4300800000,    2867200000,    1146880000,    275251200,     36700160,     2097152;
  8:  25600000000,   81920000000,  114688000000,   91750400000,   45875200000,  14680064000,   2936012800,   335544320,   16777216;
  9: 512000000000, 1843200000000, 2949120000000, 2752512000000, 1651507200000, 660602880000, 176160768000, 30198988800, 3019898880, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = (20+8*x)*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[(20 + 8*x)*P[n - 1, 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] = (20 + 8*x)*v[i - 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.