Real-rooted atlas

A038301 → reverse → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(22 + 18\,x + 11\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 198 + 162\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0198162360
132675346218710800
2479161176129622826244288000
36588452156220264627014434202952457200000
48696754355776305821794047632860194861703188646172800000
511160834354789550211206953451222576740750217545245525742334807834032000000
61403076312803580069619724238072268966882802200638132010803132648294630890434437376892160000000
7173630693611136491812723254499281885325544279445446579376703565033773841458422907483409300303234867844012073600000000

All 15 rows

Technical details
Node id
n-ec8e2539152ade57 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6bae7ffb6b8d3725

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:           198,            162;
  1:          3267,           5346,           2187;
  2:         47916,         117612,          96228,           26244;
  3:        658845,        2156220,        2646270,         1443420,          295245;
  4:       8696754,       35577630,       58217940,        47632860,        19486170,         3188646;
  5:     111608343,      547895502,     1120695345,      1222576740,       750217545,       245525742,        33480783;
  6:    1403076312,     8035800696,    19724238072,     26896688280,     22006381320,     10803132648,      2946308904,      344373768;
  7:   17363069361,   113649181272,   325449928188,    532554427944,    544657937670,    356503377384,    145842290748,    34093003032,     3486784401;
  8:  212215292190,  1562676242490,  5114213157240,   9763497845640,  11982474628740,   9803842878060,   5347550660760,  1875115166760,   383546284110,   34867844010;
  9: 2567805035499, 21009313926810, 77352474003255, 168769034188920, 241646571679590, 237252997649052, 161763407487990, 75629645059320, 23204550188655, 4219009125210, 345191655699;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (11*n+22+(9*n+18)*x)*P(n-1,x) for n >= 1, with P(0,x) = 198 + 162*x.
%t Mathematica
P[0, x_] := 198 + 162*x;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 22 + (9*n + 18)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 198 + 162*x; for(n = 1, N, my(i = n + 1); v[i] = ((11*n + 22 + (9*n + 18)*x)*v[i - 1])/(n + 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.