Real-rooted atlas

A113214 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-5\,n - 4\,n x + 4\,n x^{2} + n x^{3} + n^{2} + 5\,n^{2} x + n^{2} x^{2} + n^{3}\right)\,P_{n}(x) = \left(-15 - 18\,x + 22\,x^{2} + 10\,x^{3} + x^{4} - 4\,n + 26\,n x + n x^{2} - n x^{3} + 9\,n^{2} - 7\,n^{2} x - 2\,n^{2} x^{2} - 2\,n^{3}\right)\,P_{n-1}(x) + \left(15\,x + 48\,x^{2} + 44\,x^{3} + 12\,x^{4} + x^{5} + 5\,n + 31\,n x + 36\,n x^{2} + 11\,n x^{3} + n x^{4} + 5\,n^{2} + 6\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
297117
3202710158
4408153131188
57821020487161596
61474986494091291911852
72721111182615457161792215672
84952371471750433134114523725117168
989048931144414839116585700171630328151472
10158498352646440417385512382095752449377311153104
11279619354589241037041166448753544430151393364459341452384
12490137428127242253750329286291595179745772532282044815493711329088
138540713332679125973548793148990606541603413271270303309458206474013885632
14148051342795522571361862224357926042002188845134814560879519959046485740175343111311040
Technical details
Node id
n-6057763109e64c30 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a54a57f9f75c33a2

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 0)
  • 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:   4,    1;
  2:   9,    7,     1;
  3:  20,   27,    10,     1;
  4:  40,   81,    53,    13,     1;
  5:  78,  210,   204,    87,    16,    1;
  6: 147,  498,   649,   409,   129,   19,    1;
  7: 272, 1111,  1826,  1545,   716,  179,   22,   1;
  8: 495, 2371,  4717,  5043,  3134, 1145,  237,  25,  1;
  9: 890, 4893, 11444, 14839, 11658, 5700, 1716, 303, 28, 1;
%F Formula
The row polynomials satisfy (n^3+n^2-5*n+(5*n^2-4*n)*x+(n^2+4*n)*x^2+n*x^3)*P(n,x) = -(2*n^3-9*n^2+4*n+15+(7*n^2-26*n+18)*x+(2*n^2-n-22)*x^2+(n-10)*x^3-x^4)*P(n-1,x) + (5*n^2+5*n+(6*n^2+31*n+15)*x+(n^2+36*n+48)*x^2+(11*n+44)*x^3+(n+12)*x^4+x^5)*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[(-(2*n^3 - 9*n^2 + 4*n + 15 + (7*n^2 - 26*n + 18)*x + (2*n^2 - n - 22)*x^2 + (n - 10)*x^3 - x^4)*P[n - 1, x] + (5*n^2 + 5*n + (6*n^2 + 31*n + 15)*x + (n^2 + 36*n + 48)*x^2 + (11*n + 44)*x^3 + (n + 12)*x^4 + x^5)*D[P[n - 1, x], x])/(n^3 + n^2 - 5*n + (5*n^2 - 4*n)*x + (n^2 + 4*n)*x^2 + n*x^3)]];
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] = (-(2*n^3 - 9*n^2 + 4*n + 15 + (7*n^2 - 26*n + 18)*x + (2*n^2 - n - 22)*x^2 + (n - 10)*x^3 - x^4)*v[i - 1] + (5*n^2 + 5*n + (6*n^2 + 31*n + 15)*x + (n^2 + 36*n + 48)*x^2 + (11*n + 44)*x^3 + (n + 12)*x^4 + x^5)*deriv(v[i - 1], x))/(n^3 + n^2 - 5*n + (5*n^2 - 4*n)*x + (n^2 + 4*n)*x^2 + n*x^3)); 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. [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.