Real-rooted atlas

A038331 → x ↦ x+1 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(22 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 484 + 220\,x + 25\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
048422025729
1106487260165012519683
22342562129607260011000625531441
351536325856400266200060500068750312514348907
41133799041546089608784600026620000453750041250015625387420489
524943578883968296640270565680010248700002329250003176250024062507812510460353203
65487587353699774315520793659328003607542400010248700000186340000021175000013750000390625282429536481
71207269217792246941430912022449220992001190488992000405848520000922383000001397550000013612500007734375019531257625597484987

All 15 rows

Technical details
Node id
n-674ea42dced02dd0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-fd4e8b5768d6bd37

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,              220,               25;
  1:           10648,             7260,             1650,              125;
  2:          234256,           212960,            72600,            11000,             625;
  3:         5153632,          5856400,          2662000,           605000,           68750,            3125;
  4:       113379904,        154608960,         87846000,         26620000,         4537500,          412500,          15625;
  5:      2494357888,       3968296640,       2705656800,       1024870000,       232925000,        31762500,        2406250,         78125;
  6:     54875873536,      99774315520,      79365932800,      36075424000,     10248700000,      1863400000,      211750000,      13750000,       390625;
  7:   1207269217792,    2469414309120,    2244922099200,    1190488992000,    405848520000,     92238300000,    13975500000,    1361250000,     77343750,     1953125;
  8:  26559922791424,   60363460889600,   61735357728000,   37415368320000,  14881112400000,   4058485200000,   768652500000,   99825000000,   8507812500,   429687500,    9765625;
  9: 584318301411328, 1460795753528320, 1659995174464000, 1131814891680000, 514461314400000, 163692236400000, 37202781000000, 6039412500000, 686296875000, 51992187500, 2363281250, 48828125;
%F Formula
The row polynomials satisfy P(n,x) = (22+5*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 220*x + 25*x^2.
%t Mathematica
P[0, x_] := 484 + 220*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[(22 + 5*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] = 484 + 220*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 5*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.