Real-rooted atlas

A169803 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-39 - 81\,x + 108\,x^{2} - 21\,n + 27\,n x + 108\,n x^{2} + 45\,n^{2} + 108\,n^{2} x + 27\,n^{3}\right)\,P_{n}(x) = \left(-39 - 81\,x + 108\,x^{2} - 90\,n - 207\,n x + 486\,n x^{2} + 648\,n x^{3} + 108\,n^{2} + 432\,n^{2} x + 648\,n^{2} x^{2} + 81\,n^{3} + 162\,n^{3} x\right)\,P_{n-1}(x) + \left(-70 - 603\,x - 1899\,x^{2} - 2592\,x^{3} - 1296\,x^{4} - 120\,n - 828\,n x - 1836\,n x^{2} - 1296\,n x^{3} - 45\,n^{2} - 243\,n^{2} x - 324\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1538
29615
32033962
4388427149
57322216227484
61355105221081275
724711311593702813754
844523884239270040510177
97944899106029315283524328688
1014049792250022797212555145878183
1124651918856592775984819510935729215702
12430136972123768201312160461544325103586349
137465702512633584968004876202296354082421871598140
141289713189854761411766601376190836892224532174964324179
Technical details
Node id
n-827f95ba69e1c11f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0eb52ff90756e31e

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   5,    3;
  2:   9,    6;
  3:  20,   33,     9;
  4:  38,   84,    27;
  5:  73,  222,   162,   27;
  6: 135,  510,   522,  108;
  7: 247, 1131,  1593,  702,   81;
  8: 445, 2388,  4239, 2700,  405;
  9: 794, 4899, 10602, 9315, 2835, 243;
%F Formula
The row polynomials satisfy (27*n^3+45*n^2-21*n-39+(108*n^2+27*n-81)*x+(108*n+108)*x^2)*P(n,x) = (81*n^3+108*n^2-90*n-39+(162*n^3+432*n^2-207*n-81)*x+(648*n^2+486*n+108)*x^2+648*n*x^3)*P(n-1,x) - (45*n^2+120*n+70+(243*n^2+828*n+603)*x+(324*n^2+1836*n+1899)*x^2+(1296*n+2592)*x^3+1296*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((81*n^3 + 108*n^2 - 90*n - 39 + (162*n^3 + 432*n^2 - 207*n - 81)*x + (648*n^2 + 486*n + 108)*x^2 + 648*n*x^3)*P[n - 1, x] - (45*n^2 + 120*n + 70 + (243*n^2 + 828*n + 603)*x + (324*n^2 + 1836*n + 1899)*x^2 + (1296*n + 2592)*x^3 + 1296*x^4)*D[P[n - 1, x], x])/(27*n^3 + 45*n^2 - 21*n - 39 + (108*n^2 + 27*n - 81)*x + (108*n + 108)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((81*n^3 + 108*n^2 - 90*n - 39 + (162*n^3 + 432*n^2 - 207*n - 81)*x + (648*n^2 + 486*n + 108)*x^2 + 648*n*x^3)*v[i - 1] - (45*n^2 + 120*n + 70 + (243*n^2 + 828*n + 603)*x + (324*n^2 + 1836*n + 1899)*x^2 + (1296*n + 2592)*x^3 + 1296*x^4)*deriv(v[i - 1], x))/(27*n^3 + 45*n^2 - 21*n - 39 + (108*n^2 + 27*n - 81)*x + (108*n + 108)*x^2)); 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. [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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.