Real-rooted atlas

A290598 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 8\,x + 30\,x^{2} + 6\,n x - 33\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-6\,x^{2} + 42\,x^{3} - 18\,n x^{3}\right)\,P_{n-1}'(x) + \left(9\,x^{4}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2145
31142843
4130210280521
5152780364036408113
61802080208007280058240154001
71114456079040592800165984011065603442915
811548778234080304304018258240426025602434432088491173
91200154005852001170400012172160060860800012172160006086080002568458401
10125225200129360036867600589881600511230720021909888000383423040001704102400083033591453
11131039060260400010025400022857912003047721600022640217600084900816000013206793600005282717440002957227344571
1213745797048694802434740007498999200142480984800162835411200010584301728000352810057600004939340806400017961239296000114998537345825
13144483028857956054051228021620491200554925940800903736532160090373653216000522158885248000156647665574400019936975618560006645658539520004846887070944913
14152011544014391520111534280056213277120187377590400041223069888000587428745904000522158885248000027152262032896000740516237260800008639356101376000026582634158080000220032252718119401
1516021565202316496021659237601342872731205640065471040161144727744000310203600907200039292456114912000314339648919296000148596924943667200037149231235916800004000686440791040000114305326879744000010701533144929846003
Technical details
Node id
n-b8dd1392a9092a24 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d99ad6edcacedc3c

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,   4;
   3: 1,  14,    28;
   4: 1,  30,   210,     280;
   5: 1,  52,   780,    3640,     3640;
   6: 1,  80,  2080,   20800,    72800,     58240;
   7: 1, 114,  4560,   79040,   592800,   1659840,    1106560;
   8: 1, 154,  8778,  234080,  3043040,  18258240,   42602560,    24344320;
   9: 1, 200, 15400,  585200, 11704000, 121721600,  608608000,  1217216000,   608608000;
  10: 1, 252, 25200, 1293600, 36867600, 589881600, 5112307200, 21909888000, 38342304000, 17041024000;
%F Formula
The row polynomials satisfy P(n,x) = (1+(6*n-8)*x+(9*n^2-33*n+30)*x^2)*P(n-1,x) - (6*x^2+(18*n-42)*x^3)*d/dx P(n-1,x) + 9*x^4*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (6*n - 8)*x + (9*n^2 - 33*n + 30)*x^2)*P[n - 1, x] - (6*x^2 + (18*n - 42)*x^3)*D[P[n - 1, x], x] + 9*x^4*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (6*n - 8)*x + (9*n^2 - 33*n + 30)*x^2)*v[i - 1] - (6*x^2 + (18*n - 42)*x^3)*deriv(v[i - 1], x) + 9*x^4*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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