Real-rooted atlas

A038314 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(22 + 56\,x + 34\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 484 + 1496\,x + 1156\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0484149611563136
110648493687629639304175616
223425614481283357024345875213363369834496
351536323982352012309088019023136014699696045435424550731776
411337990410513409284061999040837017984097017993605997475968154480441630840979456
5249435788826984417152125109570432322251923840498025700480461805649536237899880064525233501441727094849536
6548758735366784653455363669880732672113432677191682191313082112027092598106112209351894456329244109625344178579390489696717311574016
712072692177921679201730201610380519786700837432783473254486775998051635213410836062525441381722503411712915166852909056353587193169408607169927664645416169448144896

All 15 rows

Technical details
Node id
n-9853514e45ba7fbd · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-94bd4dafe00b9a6d

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:             484,             1496,              1156;
  1:           10648,            49368,             76296,              39304;
  2:          234256,          1448128,           3357024,            3458752,             1336336;
  3:         5153632,         39823520,         123090880,          190231360,           146996960,            45435424;
  4:       113379904,       1051340928,        4061999040,         8370179840,          9701799360,          5997475968,          1544804416;
  5:      2494357888,      26984417152,      125109570432,       322251923840,        498025700480,        461805649536,        237899880064,         52523350144;
  6:     54875873536,     678465345536,     3669880732672,     11343267719168,      21913130821120,      27092598106112,      20935189445632,       9244109625344,       1785793904896;
  7:   1207269217792,   16792017302016,   103805197867008,    374327834732544,     867759980516352,    1341083606252544,    1381722503411712,     915166852909056,     353587193169408,      60716992766464;
  8:  26559922791424,  410471534049280,  2854642941342720,  11764589091594240,   31817865952266240,   59007678675111936,   75994737687644160,   67112235879997440,   38894591248634880,   13357738408622080,   2064377754059776;
  9: 584318301411328, 9933411123992576, 76758176867215360, 355878820020725760, 1099989080064061440, 2379976373229514752, 3678145304081977344, 4060290270739845120, 3137497027389880320, 1616286347443271680, 499579416482465792, 70188843638032384;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (22+56*x+34*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 1496*x + 1156*x^2.
%t Mathematica
P[0, x_] := 484 + 1496*x + 1156*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((22 + 56*x + 34*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 484 + 1496*x + 1156*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((22 + 56*x + 34*x^2)*v[i - 1])/(1 + 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.