Real-rooted atlas

A038330 → basis A007318 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(20 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 160\,x + 16\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
040016016576
1800048009606413824
2160000128000384005120256331776
33200000320000012800002560002560010247962624
46400000076800000384000001024000015360001228804096191102976
5128000000017920000001075200000358400000716800008601600573440163844586471424
625600000000409600000002867200000011468800000286720000045875200045875200262144065536110075314176
7512000000000921600000000737280000000344064000000103219200000206438400002752512000235929600117964802621442641807540224

All 15 rows

Technical details
Node id
n-9cce4140bcdbe78c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a8e18dd2f0709b33

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:             400,             160,              16;
  1:            8000,            4800,             960,              64;
  2:          160000,          128000,           38400,            5120,             256;
  3:         3200000,         3200000,         1280000,          256000,           25600,           1024;
  4:        64000000,        76800000,        38400000,        10240000,         1536000,         122880,          4096;
  5:      1280000000,      1792000000,      1075200000,       358400000,        71680000,        8601600,        573440,        16384;
  6:     25600000000,     40960000000,     28672000000,     11468800000,      2867200000,      458752000,      45875200,      2621440,       65536;
  7:    512000000000,    921600000000,    737280000000,    344064000000,    103219200000,    20643840000,    2752512000,    235929600,    11796480,     262144;
  8:  10240000000000,  20480000000000,  18432000000000,   9830400000000,   3440640000000,   825753600000,  137625600000,  15728640000,  1179648000,   52428800,   1048576;
  9: 204800000000000, 450560000000000, 450560000000000, 270336000000000, 108134400000000, 30277632000000, 6055526400000, 865075200000, 86507520000, 5767168000, 230686720, 4194304;
%F Formula
The row polynomials satisfy P(n,x) = (20+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 160*x + 16*x^2.
%t Mathematica
P[0, x_] := 400 + 160*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 4*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] = 400 + 160*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 4*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.