Real-rooted atlas

A121574 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-22 - 8\,x + 8\,x^{2} + 28\,n - 4\,n x - 8\,n x^{2} - 6\,n^{2} + 12\,n^{2} x\right)\,P_{n}(x) = \left(-54 - 152\,x - 136\,x^{2} - 32\,x^{3} + 158\,n + 304\,n x + 104\,n x^{2} - 48\,n^{2} - 48\,n^{2} x\right)\,P_{n-1}(x) + \left(-26 - 118\,x - 180\,x^{2} - 104\,x^{3} - 16\,x^{4} + 39\,n + 138\,n x + 132\,n x^{2} + 24\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
214822
3708824182
4310608360641342
512843432319212161609284
65100172802205613184368038461684
719682808081314961080964720010368896398546

All 15 rows

Technical details
Node id
n-40e6e342b85ff7f6 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-eea5d9915e44f7cd

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:      2;
  2:     14,       8;
  3:     70,      88,      24;
  4:    310,     608,     360,      64;
  5:   1284,    3432,    3192,    1216,     160;
  6:   5100,   17280,   22056,   13184,    3680,     384;
  7:  19682,   80808,  131496,  108096,   47200,   10368,    896;
  8:  74378,  358832,  710808,  746944,  448480,  153600,  27776,  2048;
  9: 276610, 1533544, 3582840, 4595072, 3530240, 1660800, 466816, 71680, 4608;
%F Formula
The row polynomials satisfy -(6*n^2-28*n+22-(12*n^2-4*n-8)*x+(8*n-8)*x^2)*P(n,x) = -(48*n^2-158*n+54+(48*n^2-304*n+152)*x-(104*n-136)*x^2+32*x^3)*P(n-1,x) + (39*n-26+(138*n-118)*x+(132*n-180)*x^2+(24*n-104)*x^3-16*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(48*n^2 - 158*n + 54 + (48*n^2 - 304*n + 152)*x - (104*n - 136)*x^2 + 32*x^3)*P[n - 1, x] + (39*n - 26 + (138*n - 118)*x + (132*n - 180)*x^2 + (24*n - 104)*x^3 - 16*x^4)*D[P[n - 1, x], x])/(-6*n^2 + 28*n - 22 + (12*n^2 - 4*n - 8)*x - (8*n - 8)*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(48*n^2 - 158*n + 54 + (48*n^2 - 304*n + 152)*x - (104*n - 136)*x^2 + 32*x^3)*v[i - 1] + (39*n - 26 + (138*n - 118)*x + (132*n - 180)*x^2 + (24*n - 104)*x^3 - 16*x^4)*deriv(v[i - 1], x))/(-6*n^2 + 28*n - 22 + (12*n^2 - 4*n - 8)*x - (8*n - 8)*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.