Real-rooted atlas

A200536 → basis A007318 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(6\,n + 7\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 7 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
17411
2841468416330
3756219625411464420487425
460482462442840412842380082081568128148500
5453602376005518807460406468353730201430803520050403202784375
63265922091744607824010577520122484609909654571594823505606753601291201478476850118750
7228614417309376603923041285623361863048601940775481495202178625668837260972119039202733808427392407681792877078125

All 15 rows

Technical details
Node id
n-acc13798e1eca257 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-d6c710616865b24c

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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:         0;
  1:         7,          4;
  2:        84,        146,         84,          16;
  3:       756,       2196,       2541,        1464,         420,          48;
  4:      6048,      24624,      42840,       41284,       23800,        8208,        1568,         128;
  5:     45360,     237600,     551880,      746040,      646835,      373020,      143080,       35200,        5040,         320;
  6:    326592,    2091744,    6078240,    10577520,    12248460,     9909654,     5715948,     2350560,      675360,      129120,      14784,        768;
  7:   2286144,   17309376,   60392304,   128562336,   186304860,   194077548,   149520217,    86256688,    37260972,    11903920,    2733808,     427392,     40768,      1792;
  8:  15676416,  136982016,  557819136,  1404273024,  2444057280,  3115113120,  3003741552,  2231241992,  1287317808,   576872800,  199145408,   52010112,   9947392,   1315328,   107520,    4096;
  9: 105815808, 1048080384, 4879284480, 14179878144, 28811129760, 43433578944, 50305900176, 45724352688, 33018844383, 19051813620, 8783569872, 3217302144, 924801696, 204236928, 33465600, 3833856, 274176, 9216;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (6*n+7*n*x+2*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 7 + 4*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 7*n*x + 2*n*x^2)*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] = 0; v[2] = 7 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 7*n*x + 2*n*x^2)*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.