Real-rooted atlas

A136572 → basis A013609 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13811
21896128242
31621296345630727986
41944207368294414745698304351384
529160388800207360055296007372800393216019326120
65248808398080559872001990656003981312004246732801887436801275523920
711022480205752960164602368073156608001950842880031213486080277453209601056964608098215341840
82645395205643509760526727577602809213747209364045824001997663109120266355081216020293720473606764573491208642950081920
971425670401714216089601828497162240113773156761604550926270464012135803387904021574761578496024656870375424016437913583616048704929136640855652058110080
102142770112005714053632000685686435840004875992432640002275463135232000728148203274240016181071183872000246568703754240002465687037542400014611478740992000389639433093120094121726392108800
1170711413696002074201468416002765601957888000221248156631040001179990168698880004405296629809152001174745767949107200223761098656972800029834813154263040002651983391490048000141439114212802560034288270112194560011388728893445164800
122545610893056008145954857779200119474004580761600106199115182899200063719469109739520002718697348682219520084581695292335718400193329589239624499200322215982066040832000381885608374566912000305508486699653529600148125326884680499200329167393077067776001503312213934761753600
139927882482918400344166592741171200550666548385873920053842951397729894400358953009318199296000172297444472735662080061261313590306013184001633635029074827018240032672700581496540364800484040008614763560960005163093425224144650240037549770365266506547200166887868290073362432003423340888001504870400214973646592670930764800
1441697106428257280015566919733216051200269826608709078220800287815049289683435520021106436947910118604800112567663722187299225600450270654888749196902400137225342442285469532160032019246569866609557504005692310501309619476889600758974733507949263585280073597549915922352832512004906503327728156855500800201292444214488486379520038341417945616854548480033105941575271323337779200
Technical details
Node id
n-fa5aa25df8632258 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-10cba46fcb583bff

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:          3,            8;
  2:         18,           96,           128;
  3:        162,         1296,          3456,           3072;
  4:       1944,        20736,         82944,         147456,          98304;
  5:      29160,       388800,       2073600,        5529600,        7372800,         3932160;
  6:     524880,      8398080,      55987200,      199065600,      398131200,       424673280,       188743680;
  7:   11022480,    205752960,    1646023680,     7315660800,    19508428800,     31213486080,     27745320960,     10569646080;
  8:  264539520,   5643509760,   52672757760,   280921374720,   936404582400,   1997663109120,   2663550812160,   2029372047360,    676457349120;
  9: 7142567040, 171421608960, 1828497162240, 11377315676160, 45509262704640, 121358033879040, 215747615784960, 246568703754240, 164379135836160, 48704929136640;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+8*n*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[(3*n + 8*n*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] = (3*n + 8*n*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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.