Real-rooted atlas

A059473 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + 6\,x^{2} + 2\,n - 3\,n x - 6\,n x^{2} + 3\,n^{2} x\right)\,P_{n}(x) = \left(-14 - 78\,x - 138\,x^{2} - 72\,x^{3} + 28\,n + 102\,n x + 78\,n x^{2} - 6\,n^{2} - 12\,n^{2} x\right)\,P_{n-1}(x) + \left(-4 - 32\,x - 90\,x^{2} - 102\,x^{3} - 36\,x^{4} + 4\,n + 24\,n x + 42\,n x^{2} + 18\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
166
25472126
336010086482016
42136914412312518428776
5118806854414191212441638880385632
6634324606561291464174182411275202799364964832
732928028834561021248018683136185846409517824195955262170368

All 15 rows

Technical details
Node id
n-a4bca6f0e11bfc6d · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-d440fc5b08573e95

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:       6;
  2:      54,       72;
  3:     360,     1008,       648;
  4:    2136,     9144,     12312,       5184;
  5:   11880,    68544,    141912,     124416,      38880;
  6:   63432,   460656,   1291464,    1741824,    1127520,     279936;
  7:  329280,  2883456,  10212480,   18683136,   18584640,    9517824,    1959552;
  8: 1674432, 17167680,  73529856,  170247744,  229780800,  180558720,   76422528,  13436928;
  9: 8381664, 98482176, 495004608, 1388295936, 2373274080, 2529501696, 1640145024, 591224832, 90699264;
%F Formula
The row polynomials satisfy (2*n-2+(3*n^2-3*n)*x-(6*n-6)*x^2)*P(n,x) = -(6*n^2-28*n+14+(12*n^2-102*n+78)*x-(78*n-138)*x^2+72*x^3)*P(n-1,x) + (4*n-4+(24*n-32)*x+(42*n-90)*x^2+(18*n-102)*x^3-36*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n^2 - 28*n + 14 + (12*n^2 - 102*n + 78)*x - (78*n - 138)*x^2 + 72*x^3)*P[n - 1, x] + (4*n - 4 + (24*n - 32)*x + (42*n - 90)*x^2 + (18*n - 102)*x^3 - 36*x^4)*D[P[n - 1, x], x])/(2*n - 2 + (3*n^2 - 3*n)*x - (6*n - 6)*x^2)]];
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] = 6; for(n = 2, N, my(i = n + 1); v[i] = (-(6*n^2 - 28*n + 14 + (12*n^2 - 102*n + 78)*x - (78*n - 138)*x^2 + 72*x^3)*v[i - 1] + (4*n - 4 + (24*n - 32)*x + (42*n - 90)*x^2 + (18*n - 102)*x^3 - 36*x^4)*deriv(v[i - 1], x))/(2*n - 2 + (3*n^2 - 3*n)*x - (6*n - 6)*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

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.