Real-rooted atlas

A081580 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-3 + 4\,x^{2} + 3\,n - 6\,n x - 4\,n x^{2} + 6\,n^{2} x\right)\,P_{n}(x) = \left(-51 - 240\,x - 172\,x^{2} - 32\,x^{3} + 75\,n + 318\,n x + 116\,n x^{2} - 18\,n^{2} - 66\,n^{2} x\right)\,P_{n-1}(x) + \left(-9 - 78\,x - 198\,x^{2} - 124\,x^{3} - 16\,x^{4} + 9\,n + 72\,n x + 150\,n x^{2} + 24\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
144
2363268
3168512192872
480839684416102410216
534202368048960307205120111900
614268126880381120445440189440245761181724
757232623104247872043827203399680108134411468812137488

All 15 rows

Technical details
Node id
n-5b5b86f313cb51fa · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b49b1c4f4be51f95

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:      4;
  2:     36,       32;
  3:    168,      512,      192;
  4:    808,     3968,     4416,      1024;
  5:   3420,    23680,    48960,     30720,      5120;
  6:  14268,   126880,   381120,    445440,    189440,     24576;
  7:  57232,   623104,  2478720,   4382720,   3399680,   1081344,    114688;
  8: 225936,  2909184, 14408064,  34375680,  40796160,  23150592,   5849088,   524288;
  9: 876276, 13030656, 77357952, 233607168, 380037120, 329121792, 145539072, 30408704, 2359296;
%F Formula
The row polynomials satisfy (3*n-3+(6*n^2-6*n)*x-(4*n-4)*x^2)*P(n,x) = -(18*n^2-75*n+51+(66*n^2-318*n+240)*x-(116*n-172)*x^2+32*x^3)*P(n-1,x) + (9*n-9+(72*n-78)*x+(150*n-198)*x^2+(24*n-124)*x^3-16*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(18*n^2 - 75*n + 51 + (66*n^2 - 318*n + 240)*x - (116*n - 172)*x^2 + 32*x^3)*P[n - 1, x] + (9*n - 9 + (72*n - 78)*x + (150*n - 198)*x^2 + (24*n - 124)*x^3 - 16*x^4)*D[P[n - 1, x], x])/(3*n - 3 + (6*n^2 - 6*n)*x - (4*n - 4)*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(18*n^2 - 75*n + 51 + (66*n^2 - 318*n + 240)*x - (116*n - 172)*x^2 + 32*x^3)*v[i - 1] + (9*n - 9 + (72*n - 78)*x + (150*n - 198)*x^2 + (24*n - 124)*x^3 - 16*x^4)*deriv(v[i - 1], x))/(3*n - 3 + (6*n^2 - 6*n)*x - (4*n - 4)*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.