Real-rooted atlas

A169803 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 3\,x + 4\,x^{2} + 3\,n + 7\,n x + 4\,n x^{2} + 5\,n^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(-1 + 3\,x + 4\,x^{2} + 8\,n + 25\,n x + 26\,n x^{2} + 8\,n x^{3} + 14\,n^{2} + 20\,n^{2} x + 8\,n^{2} x^{2} + 3\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-40 - 127\,x - 151\,x^{2} - 80\,x^{3} - 16\,x^{4} - 30\,n - 74\,n x - 60\,n x^{2} - 16\,n x^{3} - 5\,n^{2} - 9\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
2527
3107118
41816337
5333812184
65982344179
710517393181390
8185352231605833
93247015481852511784
10564136812425169563791
119772628272413543253318042
1216854980580833681001140716989
1328959329121028040287052542135804
14495717302247261854077501764196875243
Technical details
Node id
n-a749172fd9b39eab · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1aa2d780018369b6

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:   3,   1;
  2:   5,   2;
  3:  10,   7,   1;
  4:  18,  16,   3;
  5:  33,  38,  12,   1;
  6:  59,  82,  34,   4;
  7: 105, 173,  93,  18,  1;
  8: 185, 352, 231,  60,  5;
  9: 324, 701, 548, 185, 25, 1;
%F Formula
The row polynomials satisfy (n^3+5*n^2+3*n-1+(4*n^2+7*n+3)*x+(4*n+4)*x^2)*P(n,x) = (3*n^3+14*n^2+8*n-1+(2*n^3+20*n^2+25*n+3)*x+(8*n^2+26*n+4)*x^2+8*n*x^3)*P(n-1,x) - (5*n^2+30*n+40+(9*n^2+74*n+127)*x+(4*n^2+60*n+151)*x^2+(16*n+80)*x^3+16*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[((3*n^3 + 14*n^2 + 8*n - 1 + (2*n^3 + 20*n^2 + 25*n + 3)*x + (8*n^2 + 26*n + 4)*x^2 + 8*n*x^3)*P[n - 1, x] - (5*n^2 + 30*n + 40 + (9*n^2 + 74*n + 127)*x + (4*n^2 + 60*n + 151)*x^2 + (16*n + 80)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 + 5*n^2 + 3*n - 1 + (4*n^2 + 7*n + 3)*x + (4*n + 4)*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] = ((3*n^3 + 14*n^2 + 8*n - 1 + (2*n^3 + 20*n^2 + 25*n + 3)*x + (8*n^2 + 26*n + 4)*x^2 + 8*n*x^3)*v[i - 1] - (5*n^2 + 30*n + 40 + (9*n^2 + 74*n + 127)*x + (4*n^2 + 60*n + 151)*x^2 + (16*n + 80)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 + 5*n^2 + 3*n - 1 + (4*n^2 + 7*n + 3)*x + (4*n + 4)*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.