Real-rooted atlas

A059364 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(1 + 6\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17411
29110832231
31729314419043847161
44322510626497904400646144293601
5133997541586845160304320006499174412288014973651
649579075186030708290739664242249664113496064283484162949120913392711
72131900225938753454417710665376185574461441166332134443968716809205678088257536064850882481
81044631110255282099998561168223708832147605615308811653410224645886729953281858077163523350436249626424115205252921480961
95745471106375328122219889808326786398057612323914193779211723177774668874329291551744314116522311688531817725952135148968345695126814720478015854767451
1035047373748887522313643855827806391828582374336108483022174283521208070412005964892233578945239044889282448171008177690697052979242371357972889659862323036160380507258880048279601331512551
11234817404117546251649222582835567605264325479847228321008076706192170368128673247361084390411495159602157260807334096713865093123341811947450204161065727165451468802265417315149414428888820770406401674231939072005359035747797893161
1217141670500580876251316644839446386548046345844400495491216988583618552951328001423191524708157826561456778238290685112321087156721015700316165959889144293893734423820505655556702208676924503422612275212982887028956856321508882328531763208036313307545600648443325483545072481
131354191969545889223751129286109765665929420434597702415126481102410219794495394081034432163838478616697937354241890914401097883308544016163784935135595081728103615273732738176942084980962301821804216320177343665179281221222445456554930851754803279431182945976582144848105685966127104041788829199237120084946075638344404495011
14115106317411400584018751035727943595385836537004326480691997348009845601112057245460967717344064196493555998400100543923225247727211867092614461442432833784101340234731520178590178309586782858444810036273285581871492628484296760043044100741529601379505241916214754672643220732131168499217203251690360780458966384645104596889603827302402340174435157278720011977396665006561033796551
Technical details
Node id
n-64929b67dfd5fcf0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-4db497742e941c24

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:             7,              4;
  2:            91,            108,             32;
  3:          1729,           3144,           1904,             384;
  4:         43225,         106264,          97904,           40064,            6144;
  5:       1339975,        4158684,        5160304,         3200064,          991744,         122880;
  6:      49579075,      186030708,      290739664,       242249664,       113496064,       28348416,        2949120;
  7:    2131900225,     9387534544,    17710665376,     18557446144,     11663321344,     4396871680,      920567808,      82575360;
  8:  104463111025,   528209999856,  1168223708832,   1476056153088,   1165341022464,   588672995328,   185807716352,   33504362496,    2642411520;
  9: 5745471106375, 32812221988980, 83267863980576, 123239141937792, 117231777746688, 74329291551744, 31411652231168, 8531817725952, 1351489683456, 95126814720;
%F Formula
The row polynomials satisfy P(n,x) = (6*n+1+4*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[(6*n + 1 + 4*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] = (6*n + 1 + 4*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

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.