Real-rooted atlas

A059473 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(3\,n\right)\,P_{n}(x) = \left(18\,x^{2} + 8\,x^{3} + 18\,n - 3\,n x - 30\,n x^{2} - 12\,n x^{3} + 15\,n^{2} x + 12\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(60\,x^{2} + 66\,x^{3} + 16\,x^{4} - 18\,n x - 48\,n x^{2} - 36\,n x^{3} - 8\,n x^{4}\right)\,P_{n-1}'(x) + \left(18\,x^{2} + 42\,x^{3} + 24\,x^{4} + 4\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
2365418108
3216504360801160
4129641044572213635612464
57776311044730434272118801584133920
6466562255044354564304882303286343270481438912
7279936158630437169284670784340416014417283292803136015460480

All 15 rows

Technical details
Node id
n-24292f95f290f636 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-fbbb21d417a019c4

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:        6,        4;
  2:       36,       54,        18;
  3:      216,      504,       360,        80;
  4:     1296,     4104,      4572,      2136,       356;
  5:     7776,    31104,     47304,     34272,     11880,      1584;
  6:    46656,   225504,    435456,    430488,    230328,     63432,      7048;
  7:   279936,  1586304,   3716928,   4670784,   3404160,   1441728,    329280,    31360;
  8:  1679616, 10917504,  30093120,  45956160,  42561936,  24509952,   8583840,  1674432,  139536;
  9: 10077696, 73903104, 234306432, 421583616, 474654816, 347073984, 165001536, 49241088, 8381664, 620864;
%F Formula
The row polynomials satisfy 3*n*P(n,x) = (18*n+(15*n^2-3*n)*x+(12*n^2-30*n+18)*x^2+(4*n^2-12*n+8)*x^3)*P(n-1,x) - (18*n*x+(48*n-60)*x^2+(36*n-66)*x^3+(8*n-16)*x^4)*d/dx P(n-1,x) + (18*x^2+42*x^3+24*x^4+4*x^5)*d^2/dx^2 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[((18*n + (15*n^2 - 3*n)*x + (12*n^2 - 30*n + 18)*x^2 + (4*n^2 - 12*n + 8)*x^3)*P[n - 1, x] - (18*n*x + (48*n - 60)*x^2 + (36*n - 66)*x^3 + (8*n - 16)*x^4)*D[P[n - 1, x], x] + (18*x^2 + 42*x^3 + 24*x^4 + 4*x^5)*D[P[n - 1, x], {x, 2}])/(3*n)]];
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] = ((18*n + (15*n^2 - 3*n)*x + (12*n^2 - 30*n + 18)*x^2 + (4*n^2 - 12*n + 8)*x^3)*v[i - 1] - (18*n*x + (48*n - 60)*x^2 + (36*n - 66)*x^3 + (8*n - 16)*x^4)*deriv(v[i - 1], x) + (18*x^2 + 42*x^3 + 24*x^4 + 4*x^5)*deriv(deriv(v[i - 1], x), x))/(3*n)); 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