Real-rooted atlas

A038331 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5 + 17\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 25 + 170\,x + 289\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
025170289484
112512754335491310648
26258500433509826083521234256
33125531253612501228250208802514198575153632
415625318750270937512282500313203754259571024137569113379904
5781251859375189656251074718753654043757454249258448149154103386732494357888
639062510625000126437500859775000365404375099389990001689629830016413546920697575744154875873536
71953125597656258128125006448312500328863937501118137387502534444745003693048057003139090848451185878764971207269217792

All 15 rows

Technical details
Node id
n-ac4003e8d2a030bb · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-75fb8f2b619cd489

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:       25,        170,         289;
  1:      125,       1275,        4335,         4913;
  2:      625,       8500,       43350,        98260,         83521;
  3:     3125,      53125,      361250,      1228250,       2088025,        1419857;
  4:    15625,     318750,     2709375,     12282500,      31320375,       42595710,       24137569;
  5:    78125,    1859375,    18965625,    107471875,     365404375,      745424925,      844814915,      410338673;
  6:   390625,   10625000,   126437500,    859775000,    3654043750,     9938999000,    16896298300,    16413546920,      6975757441;
  7:  1953125,   59765625,   812812500,   6448312500,   32886393750,   111813738750,   253444474500,   369304805700,    313909084845,    118587876497;
  8:  9765625,  332031250,  5080078125,  46059375000,  274053281250,  1118137387500,  3168055931250,  6155080095000,   7847727121125,   5929393824850,   2015993900449;
  9: 48828125, 1826171875, 31044921875, 316658203125, 2153275781250, 10249592718750, 34848615243750, 84632351306250, 143874997220625, 163058330183375, 110879664524695, 34271896307633;
%F Formula
The row polynomials satisfy P(n,x) = (5+17*x)*P(n-1,x) for n >= 1, with P(0,x) = 25 + 170*x + 289*x^2.
%t Mathematica
P[0, x_] := 25 + 170*x + 289*x^2;
P[n_, x_] := P[n, x] = Expand[(5 + 17*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] = 25 + 170*x + 289*x^2; for(n = 1, N, my(i = n + 1); v[i] = (5 + 17*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

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.