Real-rooted atlas

A225471 → 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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1516
24514160
3585227271840
49945444468644115120
5208845103269188501610651332640
6522112527905705745195910032359018648640
71514126258614765519451621228841915291558451191259459200
849966166252994285240728051148949694487334614345800977215218821612800
918487481512511578517050529932177716424192072436635016620129214707364153961891335221286400
1075798674201254932066805830134300445686120385092740019262277530119164794049131138133860023145230114079294028800
11341094033905625229522873682475653672673645751051629618986110706534162507288643483034025491501093681382380125334952751647647525324800
12167136076613756251158771484434690034325189745466505806657806677646297831358611146420887229202396113431808761587912225994263402138046970324132382376266240000
13885821206052908125630862494411761325193511220495319350342078053499381423918516400731647309009015900871173414899114607039755025162073928385143912740365107906414237711748648318376960000
1450491808745015763125368449833875233036501166100206264496427521433561254417934442575632401916420212153203030708129412974739408540915746809355487218861146820234576954582281024243310166884856314341101421602465863680000
Technical details
Node id
n-8ea111b9cf561069 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-3c30cec442f32d06

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,            1;
  2:           45,           14,           1;
  3:          585,          227,          27,          1;
  4:         9945,         4444,         686,         44,         1;
  5:       208845,       103269,       18850,       1610,        65,        1;
  6:      5221125,      2790570,      574519,      59100,      3235,       90,      1;
  7:    151412625,     86147655,    19451621,    2288419,    152915,     5845,    119,     1;
  8:   4996616625,   2994285240,   728051148,   94969448,   7334614,   345800,   9772,   152,   1;
  9: 184874815125, 115785170505, 29932177716, 4241920724, 366350166, 20129214, 707364, 15396, 189, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+1+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[(4*n + 1 + 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] = (4*n + 1 + 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.