Real-rooted atlas

A257615 → D

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,n\right)\,P_{n}(x) = \left(-105 + 30\,x + 3\,x^{2} - 70\,n - 40\,n x + 2\,n x^{2} - 40\,n^{2} x - 12\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-30\,x + 24\,x^{2} + 6\,x^{3} - 20\,n x + 4\,n x^{2} + 16\,n x^{3} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(22 + 2\,x + 24\,n + 8\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 5\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
155
27050120
371514303752520
463802574019140250053760
552785372020558030211140156251209600
6416370472059012276660944118020818509375029030400
731802155497641022709476530279302013744102519081290546875745113600

All 15 rows

Technical details
Node id
n-2e0e4d5e661bfdfc · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-de1cf4a5947ef456

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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:         0;
  1:         5;
  2:        70,         50;
  3:       715,       1430,         375;
  4:      6380,      25740,       19140,         2500;
  5:     52785,     372020,      558030,       211140,        15625;
  6:    416370,    4720590,    12276660,      9441180,      2081850,        93750;
  7:   3180215,   54976410,   227094765,    302793020,    137441025,     19081290,      546875;
  8:  23745880,  602915000,  3735011640,   7872618520,   6225019400,   1808745000,   166221160,    3125000;
  9: 174424285, 6328578440, 56459460540, 177044273560, 221305341950, 112918921080, 22150024540, 1395394280, 17578125;
%F Formula
The row polynomials satisfy (2*n+1)*P(n,x) = -(70*n+105+(40*n^2+40*n-30)*x+(8*n^3+12*n^2-2*n-3)*x^2)*P(n-2,x) - ((20*n+30)*x+(8*n^2-4*n-24)*x^2-(8*n^2+16*n+6)*x^3)*d/dx P(n-2,x) + (24*n+22+(8*n^2+8*n+2)*x)*P(n-1,x) + ((4*n+2)*x-(4*n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 5.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 5;
P[n_, x_] := P[n, x] = Expand[Together[(-(70*n + 105 + (40*n^2 + 40*n - 30)*x + (8*n^3 + 12*n^2 - 2*n - 3)*x^2)*P[n - 2, x] - ((20*n + 30)*x + (8*n^2 - 4*n - 24)*x^2 - (8*n^2 + 16*n + 6)*x^3)*D[P[n - 2, x], x] + (24*n + 22 + (8*n^2 + 8*n + 2)*x)*P[n - 1, x] + ((4*n + 2)*x - (4*n + 2)*x^2)*D[P[n - 1, x], x])/(2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 5; for(n = 2, N, my(i = n + 1); v[i] = (-(70*n + 105 + (40*n^2 + 40*n - 30)*x + (8*n^3 + 12*n^2 - 2*n - 3)*x^2)*v[i - 2] - ((20*n + 30)*x + (8*n^2 - 4*n - 24)*x^2 - (8*n^2 + 16*n + 6)*x^3)*deriv(v[i - 2], x) + (24*n + 22 + (8*n^2 + 8*n + 2)*x)*v[i - 1] + ((4*n + 2)*x - (4*n + 2)*x^2)*deriv(v[i - 1], x))/(2*n + 1)); 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.