Real-rooted atlas

A038301 → reverse → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(20 + 9\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 360\,x + 81\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
040036081841
1800010800486072924389
2160000288000194400583206561707281
332000007200000648000029160006561005904920511149
464000000172800000194400000116640000393660007085880531441594823321
51280000000403200000054432000004082400000183708000049601160074401740478296917249876309
625600000000921600000001451520000001306368000007348320000026453952000595213920076527504043046721500246412961
75120000000002073600000000373248000000039191040000002645395200000119042784000035712835200068874753600774840978038742048914507145975869

All 15 rows

Technical details
Node id
n-879e9349060a8fc9 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-9eb007f6301418ee

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,              360,               81;
  1:            8000,            10800,             4860,              729;
  2:          160000,           288000,           194400,            58320,             6561;
  3:         3200000,          7200000,          6480000,          2916000,           656100,            59049;
  4:        64000000,        172800000,        194400000,        116640000,         39366000,          7085880,          531441;
  5:      1280000000,       4032000000,       5443200000,       4082400000,       1837080000,        496011600,        74401740,         4782969;
  6:     25600000000,      92160000000,     145152000000,     130636800000,      73483200000,      26453952000,      5952139200,       765275040,       43046721;
  7:    512000000000,    2073600000000,    3732480000000,    3919104000000,    2645395200000,    1190427840000,    357128352000,     68874753600,     7748409780,     387420489;
  8:  10240000000000,   46080000000000,   93312000000000,  111974400000000,   88179840000000,   47617113600000,  17856417600000,   4591650240000,   774840978000,   77484097800,   3486784401;
  9: 204800000000000, 1013760000000000, 2280960000000000, 3079296000000000, 2771366400000000, 1745960832000000, 785682374400000, 252540763200000, 56821671720000, 8523250758000, 767092568220, 31381059609;
%F Formula
The row polynomials satisfy P(n,x) = (20+9*x)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 360*x + 81*x^2.
%t Mathematica
P[0, x_] := 400 + 360*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[(20 + 9*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 + 360*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = (20 + 9*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.