Real-rooted atlas

A196347 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,n x\right)\,P_{n}(x) = \left(2\,n + 8\,n x + 8\,n x^{2} + 4\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
12810
286496168
348576172815363888
43846144276484915230720114048
5384076800460800122880015360007372804043520
6460801105920829440029491200552960005308416020643840167961600
7645120180633601625702407225344001806336000260112384020230963206606028807994972160
810321920330301440346816512018496880640578027520001109812838401294781644808455716864023781703680428906741760
91857945606688604160802632499204994157772801872809164800449474199552069918208819206849130659840385263599616095126814720025598960271360
10371589120014863564800020065812480001426902220800062426972160000179789679820800349591044096000456608710656000385263599616000190253629440000418557984768001682217389260800
11817496064003596982681600539547402240004316379217920002158189608960000725151708610560016920206534246400276248269946880003107793036902400023020689162240000101291032313856002009078326886400120680812707840000
121961990553600941755465728001553896518451200138124134973440007769482592256000029834813154263040081216991364382720015911900348940288002237610986569728000220998615957504000014585908653195264005786145581432832001044720729980928009384139996161638400
1351011754393600265261122846720047747002112409600466859576210227200291787235131392000012605208557676134400392162044016590848008963703863236362240015126250269211361280018674383048409088000016433457082599997440097785860326214860800353115606733553664005850436087893196800786182395233986150400
141428329123020800799864308891648001559735402338713600166371776249462784001143805961715056640005490268616232271872001921594015681295155200501967416341236285440098824835092180893696001464071630995272499200016104787940947997491200127773524159587418112006921065891977651814400229337094645413314560035102616527359180800070593757144458618470400
Technical details
Node id
n-def320549df20ad6 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7cabb77b45806852

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:         2,          8;
  2:         8,         64,          96;
  3:        48,        576,        1728,         1536;
  4:       384,       6144,       27648,        49152,         30720;
  5:      3840,      76800,      460800,      1228800,       1536000,        737280;
  6:     46080,    1105920,     8294400,     29491200,      55296000,      53084160,      20643840;
  7:    645120,   18063360,   162570240,    722534400,    1806336000,    2601123840,    2023096320,     660602880;
  8:  10321920,  330301440,  3468165120,  18496880640,   57802752000,  110981283840,  129478164480,   84557168640,   23781703680;
  9: 185794560, 6688604160, 80263249920, 499415777280, 1872809164800, 4494741995520, 6991820881920, 6849130659840, 3852635996160, 951268147200;
%F Formula
The row polynomials satisfy (1+2*n*x)*P(n,x) = (2*n+(4*n^2+8*n)*x+(8*n^2+8*n)*x^2)*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[Together[((2*n + (4*n^2 + 8*n)*x + (8*n^2 + 8*n)*x^2)*P[n - 1, x])/(1 + 2*n*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] = ((2*n + (4*n^2 + 8*n)*x + (8*n^2 + 8*n)*x^2)*v[i - 1])/(1 + 2*n*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

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.