Real-rooted atlas

A038281 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(14 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(187 + 84\,x + 9\,x^{2} + 33\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 187 + 84\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0187849280
124201683378274508
230613290401009815128171344
33806664591952178005049056702431114064
446704796851988413275513068002272052041272917210368
5566899529808005971945874289292856860700954261714422187263533760
66820509851360558848117696070857556699217357571032931360381704424494465614005713152
781456374781841537659518367544448105926463723885077196937308834148191120147228848266861968360507351296

All 15 rows

Technical details
Node id
n-5de516ea0ecc91dc · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bb24aeb74e5c1fb8

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:           187,            84,             9;
  1:          2420,          1683,           378,            27;
  2:         30613,         29040,         10098,          1512,            81;
  3:        380666,        459195,        217800,         50490,          5670,          243;
  4:       4670479,       6851988,       4132755,       1306800,        227205,        20412,          729;
  5:      56689952,      98080059,      71945874,      28929285,       6860700,       954261,        71442,        2187;
  6:     682050985,    1360558848,    1176960708,     575566992,     173575710,     32931360,      3817044,      244944,       6561;
  7:    8145637478,   18415376595,   18367544448,   10592646372,    3885077196,    937308834,    148191120,    14722884,     826686,     19683;
  8:   96675855331,  244369124340,  276230648925,  183675444480,   79444847790,  23310463176,   4686544170,   635104800,   55210815,   2755620,   59049;
  9: 1141246682444, 3190303225923, 4032090551610, 3038537138175, 1515322416960, 524335995414, 128207547468, 22093708230, 2619807300, 202439655, 9093546, 177147;
%F Formula
The row polynomials satisfy (3*n+14+3*x)*P(n,x) = (33*n+187+(9*n+84)*x+9*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 187 + 84*x + 9*x^2.
%t Mathematica
P[0, x_] := 187 + 84*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((33*n + 187 + (9*n + 84)*x + 9*x^2)*P[n - 1, x])/(3*n + 14 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 187 + 84*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((33*n + 187 + (9*n + 84)*x + 9*x^2)*v[i - 1])/(3*n + 14 + 3*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.