Real-rooted atlas

A128813 → basis A007318 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
22116340
327333413518760
4573397446175172818023560
51777233880593375851461933150027001083760
6764208920418720226653941337558444245537776005670069360640
74355990731377845532186365161813970393206267158731682106842500470015876005895654400
83179873232911626429046418564900726616907532860896043674249348411513607880927724101606400057153600642626329600
92893684641939120109933870292212595273208623740060230298163483641440197492510114965228501623408444710357158050923857600257191200087397180825600
103211989952552291492372921266864311658538954814138520740858278083120371732652499173082865068216066572485830639917530438932000030243244630500308629440000014145516000014507932017049600
11427194663689454572196847322153380265130024696135157777180204891252757706692463301074176954361447448957922142027760970997736276725332442778233415592441230976500240653130081300022250896668000093360405600002887078471392870400
126706956219924436749378226213345858157289767679684211834301715274340942252252649616107727197863608817512067593869701795780865855137346536252131263753075089441693447693772578630332552159403583024380800022264334923217400018821457768960000728211163680000678463440777324544000
1312273729882461719250677531872720242328022383213167123635549660033593684063243695439156331543376791029539954047161367417285316101703027892303300055584476830050110819498772671581933873380811336520327026732099184186158997712623218580099346167437000023704871551847514000184601529992880000066267215894880000185898982772986925056000
1425897570051994227618913717180993077369792648548485288672944335542269058964100116282423706480502121601302076931311314141809904612307067778271441108840384871784461310231399319855647013792993974044595998081939559291188406129461185609130536141828551703552143347405087360890324360245108322109156343040002878520741228965770000207813989046343680000695805766896240000058744078556263868317696000

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(1 + n + \frac{1}{2}\,n x + n^{2} + \frac{1}{2}\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
Technical details
Node id
n-2accfa5d21a32af7 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-5cb59165f26847ca)
Fingerprint
f6-f9525b8f12f7179b

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,              1;
  2:            21,             16,              3;
  3:           273,            334,            135,             18;
  4:          5733,           9744,           6175,           1728,            180;
  5:        177723,         388059,         337585,         146193,          31500,          2700;
  6:       7642089,       20418720,       22665394,       13375584,        4424553,        777600,         56700;
  7:     435599073,     1377845532,     1863651618,     1397039320,      626715873,     168210684,      25004700,      1587600;
  8:   31798732329,   116264290464,   185649007266,   169075328608,    96043674249,   34841151360,    7880927724,   1016064000,    57153600;
  9: 2893684641939, 12010993387029, 22125952732086, 23740060230298, 16348364144019, 7492510114965, 2285016234084, 447103571580, 50923857600, 2571912000;
%F Formula
The row polynomials satisfy 2*P(n,x) = (2*n^2+2*n+2+(n^2+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[Together[((2*n^2 + 2*n + 2 + (n^2 + n)*x)*P[n - 1, x])/(2)]];
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^2 + 2*n + 2 + (n^2 + n)*x)*v[i - 1])/(2)); 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.