Real-rooted atlas

A104684 → D → D

nonnegative coefficients

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

Recurrence

\(\left(9 - 6\,n + n^{2}\right)\,P_{n}(x) = \left(20 + 19\,x - 18\,n - 16\,n x + 4\,n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,x + 4\,x^{2} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12Pn(1)
10
20
322
424630
518012012312
611201260360202760
763001008050408403022290
833264693005040015120168042169806
9168168432432415800184800378003024561242080
10823680252252030270241801800554400831605040728817696
1139382201400256020180160151351206306300144144016632079209061178130
1218475600748261801260230401143542406054048018918900336336030888011880110416822670
1385357272387987600748261800798145920514594080205837632504504007207200540540171601322798399736
1438939846419632172564267863600523783260039907296001955457504617512896122522400144144009009002402415618559873800
1517577014009734961600235586070723272028760028808079300167610643206518191680167610643227567540027227200144144032760182121839376386
Technical details
Node id
n-4efe71505c1f5edb · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6832a290ef8b97c2

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -3)
  • 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:      0;
   2:      0;
   3:      2;
   4:     24,       6;
   5:    180,     120,      12;
   6:   1120,    1260,     360,      20;
   7:   6300,   10080,    5040,     840,     30;
   8:  33264,   69300,   50400,   15120,   1680,    42;
   9: 168168,  432432,  415800,  184800,  37800,  3024,   56;
  10: 823680, 2522520, 3027024, 1801800, 554400, 83160, 5040, 72;
%F Formula
The row polynomials satisfy (n^2-6*n+9)*P(n,x) = (4*n^2-18*n+20+(3*n^2-16*n+19)*x)*P(n-1,x) - ((2*n-4)*x+(2*n-4)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 18*n + 20 + (3*n^2 - 16*n + 19)*x)*P[n - 1, x] - ((2*n - 4)*x + (2*n - 4)*x^2)*D[P[n - 1, x], x])/(n^2 - 6*n + 9)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 4, N, my(i = n); v[i] = ((4*n^2 - 18*n + 20 + (3*n^2 - 16*n + 19)*x)*v[i - 1] - ((2*n - 4)*x + (2*n - 4)*x^2)*deriv(v[i - 1], x))/(n^2 - 6*n + 9)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023