Real-rooted atlas

A121314 → D+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 12\,x + 39\,x^{2} - 12\,x^{3} - 10\,n + 32\,n x - 50\,n x^{2} + 8\,n x^{3} + 4\,n^{2} - 28\,n^{2} x + 16\,n^{2} x^{2} + 8\,n^{3} x\right)\,P_{n}(x) = \left(6 + 30\,x + 41\,x^{2} + 55\,x^{3} - 28\,x^{4} - 10\,n - 34\,n x - 16\,n x^{2} - 102\,n x^{3} + 24\,n x^{4} + 4\,n^{2} - 4\,n^{2} x - 44\,n^{2} x^{2} + 48\,n^{2} x^{3} + 8\,n^{3} x + 24\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(2\,x + 10\,x^{2} + 13\,x^{3} + 16\,x^{4} - 16\,x^{5} - 2\,n x - 6\,n x^{2} - 32\,n x^{4} - 4\,n^{2} x^{2} - 16\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
21315
3176115
41112310146
51155255151139
611993168110211413
71231463814341962811210

All 15 rows

Technical details
Node id
n-45d24d88ef8e58a8 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-19dd6beff63092aa

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: 1,  7,   6,    1;
  4: 1, 11,  23,   10,    1;
  5: 1, 15,  52,   55,   15,    1;
  6: 1, 19,  93,  168,  110,   21,    1;
  7: 1, 23, 146,  381,  434,  196,   28,   1;
  8: 1, 27, 211,  726, 1215,  966,  322,  36,  1;
  9: 1, 31, 288, 1235, 2761, 3267, 1932, 498, 45, 1;
%F Formula
The row polynomials satisfy (4*n^2-10*n+6+(8*n^3-28*n^2+32*n-12)*x+(16*n^2-50*n+39)*x^2+(8*n-12)*x^3)*P(n,x) = (4*n^2-10*n+6+(8*n^3-4*n^2-34*n+30)*x+(24*n^3-44*n^2-16*n+41)*x^2+(48*n^2-102*n+55)*x^3+(24*n-28)*x^4)*P(n-1,x) - ((2*n-2)*x+(4*n^2+6*n-10)*x^2+(16*n^2-13)*x^3+(32*n-16)*x^4+16*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^2 - 10*n + 6 + (8*n^3 - 4*n^2 - 34*n + 30)*x + (24*n^3 - 44*n^2 - 16*n + 41)*x^2 + (48*n^2 - 102*n + 55)*x^3 + (24*n - 28)*x^4)*P[n - 1, x] - ((2*n - 2)*x + (4*n^2 + 6*n - 10)*x^2 + (16*n^2 - 13)*x^3 + (32*n - 16)*x^4 + 16*x^5)*D[P[n - 1, x], x])/(4*n^2 - 10*n + 6 + (8*n^3 - 28*n^2 + 32*n - 12)*x + (16*n^2 - 50*n + 39)*x^2 + (8*n - 12)*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^2 - 10*n + 6 + (8*n^3 - 4*n^2 - 34*n + 30)*x + (24*n^3 - 44*n^2 - 16*n + 41)*x^2 + (48*n^2 - 102*n + 55)*x^3 + (24*n - 28)*x^4)*v[i - 1] - ((2*n - 2)*x + (4*n^2 + 6*n - 10)*x^2 + (16*n^2 - 13)*x^3 + (32*n - 16)*x^4 + 16*x^5)*deriv(v[i - 1], x))/(4*n^2 - 10*n + 6 + (8*n^3 - 28*n^2 + 32*n - 12)*x + (16*n^2 - 50*n + 39)*x^2 + (8*n - 12)*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.