Real-rooted atlas

A088996 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-1 + 6\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
151217
255252288595
39356264139681036831535
4215051889526219369089284976642238985
5623645676990829373264636750726896793629859840199269665
6218275752818492201515497616434350252869984829446010785792214990848021321854155
789493057513389334320858107367363053854033926517920130568343147847685930522542081805923123202665231769375
842061737025715212048240531848072131222590944686080599512233392641017848281374721079676752855046542085508300817336861982720381128143020625
9222927206232542448906155420359122379439456177171598626393656172368630776321186932801192652816715048228978688151278142502338567984306034049024187238109413376061361631026320625
101315270516771752771998110648780262820891255783041476259287223069445440228932732526081374358776272980992241050720694092595228983468282412400642286411766036955136106858720863977472022468573129605120010983731953711391875
11854925835901637519754144801355780020742395437682287201306492113152628748854848110654105936128161144342369813108736338098326919193677824506579495151882829824531198546120245772288371264521678463434752155658084074693591040296585165310787584002163795194881144199375
12606997343490162625152565360126609618001757169761214965623201226299894640991347328577556449938130686336019339376242447985522688472097665125158387834888465330323214757052416011066254407640857732710410285237168048629507686464513815091001979568128245205187804624689561604270826380475341209600465215966899446002865625
13467387954487425221251269444858559359428100159102267793303453394401218369401623097974661766360212500163552786618882390120032571978755931136665209471528560532786790413883027924827833780584448217269311980984814559559682518298949531718278394675221012533744163014458736640119522351002919189222522884154054561048746430300160666248915354153228697600108395320287570918667690625
1438793200222456293363751132160408958155762493001533815558922391047094320127853841323992112067185287325836969862553410325427230523153270622192355748147295366402684079767320946688022698432299286918988703170564135683980813250035982532608574031256939187105566818304059747825359788169077781954564522141182343615699614695424235276202541608833265329766475317982623058411827232768011192981777949774242119680027207225392180300585590346875
Technical details
Node id
n-447b8fa6e8cf1eaa · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-47aff865bb54210c

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:             5,             12;
  2:            55,            252,             288;
  3:           935,           6264,           13968,            10368;
  4:         21505,         188952,          621936,           908928,           497664;
  5:        623645,        6769908,        29373264,         63675072,         68967936,          29859840;
  6:      21827575,      281849220,      1515497616,       4343502528,       6998482944,        6010785792,        2149908480;
  7:     894930575,    13389334320,     85810736736,     305385403392,     651792013056,      834314784768,      593052254208,      180592312320;
  8:   42061737025,   715212048240,   5318480721312,   22590944686080,   59951223339264,   101784828137472,   107967675285504,    65420855083008,   17336861982720;
  9: 2229272062325, 42448906155420, 359122379439456, 1771715986263936, 5617236863077632, 11869328011926528, 16715048228978688, 15127814250233856, 7984306034049024, 1872381094133760;
%F Formula
The row polynomials satisfy P(n,x) = (6*n-1+12*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 + 12*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 + 12*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.