Real-rooted atlas

A107131 → D+1

nonnegative coefficients

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

Recurrence

\(\left(12\,x - 12\,x^{2} + 2\,x^{3} + 2\,n - 4\,n x - 2\,n x^{2} + n x^{3} - n^{2} - 3\,n^{2} x + 2\,n^{2} x^{2} - n^{3} + n^{3} x\right)\,P_{n}(x) = \left(24\,x - 24\,x^{2} + 4\,x^{3} + 12\,n - 38\,n x + 36\,n x^{2} - 19\,n x^{3} + 3\,n x^{4} - 16\,n^{2} + 27\,n^{2} x - 21\,n^{2} x^{2} + 6\,n^{2} x^{3} + 4\,n^{3} - 7\,n^{3} x + 3\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(16\,x^{2} - 36\,x^{3} + 16\,x^{4} - 2\,x^{5} + 8\,n x - 30\,n x^{2} + 23\,n x^{3} - 4\,n x^{4} - 8\,n^{2} x + 10\,n^{2} x^{2} - 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
21315
3066113
4042010135
500305015196
60015125105211267
7000140385196281750
8000567149803363612123
90000630264621845404516046
10000021038227980441082555117303
1100000277216632207908250121066149721
1200000792195365821248510145201716781143365
130000001201297812174174103818243102366911414584
1400000030039652539039046246220720739039318510511201917
Technical details
Node id
n-c31232a2bf3922ed · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-6773f8dea27f092f

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: 1, 1;
  2: 1, 3,  1;
  3: 0, 6,  6,   1;
  4: 0, 4, 20,  10,   1;
  5: 0, 0, 30,  50,  15,    1;
  6: 0, 0, 15, 125, 105,   21,    1;
  7: 0, 0,  0, 140, 385,  196,   28,   1;
  8: 0, 0,  0,  56, 714,  980,  336,  36,  1;
  9: 0, 0,  0,   0, 630, 2646, 2184, 540, 45, 1;
%F Formula
The row polynomials satisfy -(n^3+n^2-2*n-(n^3-3*n^2-4*n+12)*x-(2*n^2-2*n-12)*x^2-(n+2)*x^3)*P(n,x) = (4*n^3-16*n^2+12*n-(7*n^3-27*n^2+38*n-24)*x+(3*n^3-21*n^2+36*n-24)*x^2+(6*n^2-19*n+4)*x^3+3*n*x^4)*P(n-1,x) - ((8*n^2-8*n)*x-(10*n^2-30*n+16)*x^2+(2*n^2-23*n+36)*x^3+(4*n-16)*x^4+2*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[((4*n^3 - 16*n^2 + 12*n - (7*n^3 - 27*n^2 + 38*n - 24)*x + (3*n^3 - 21*n^2 + 36*n - 24)*x^2 + (6*n^2 - 19*n + 4)*x^3 + 3*n*x^4)*P[n - 1, x] - ((8*n^2 - 8*n)*x - (10*n^2 - 30*n + 16)*x^2 + (2*n^2 - 23*n + 36)*x^3 + (4*n - 16)*x^4 + 2*x^5)*D[P[n - 1, x], x])/(-n^3 - n^2 + 2*n + (n^3 - 3*n^2 - 4*n + 12)*x + (2*n^2 - 2*n - 12)*x^2 + (n + 2)*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] = ((4*n^3 - 16*n^2 + 12*n - (7*n^3 - 27*n^2 + 38*n - 24)*x + (3*n^3 - 21*n^2 + 36*n - 24)*x^2 + (6*n^2 - 19*n + 4)*x^3 + 3*n*x^4)*v[i - 1] - ((8*n^2 - 8*n)*x - (10*n^2 - 30*n + 16)*x^2 + (2*n^2 - 23*n + 36)*x^3 + (4*n - 16)*x^4 + 2*x^5)*deriv(v[i - 1], x))/(-n^3 - n^2 + 2*n + (n^3 - 3*n^2 - 4*n + 12)*x + (2*n^2 - 2*n - 12)*x^2 + (n + 2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.