Real-rooted atlas

A249250 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-24\,x - 36\,x^{2} - 13\,x^{3} - x^{4} - 12\,n - 21\,n x + 2\,n x^{2} + 10\,n x^{3} + n x^{4} + 12\,n^{2} x + 12\,n^{2} x^{2} + n^{2} x^{3} + 3\,n^{3} x\right)\,P_{n}(x) = \left(-24\,x - 36\,x^{2} - 13\,x^{3} - x^{4} - 12\,n - 102\,n x - 148\,n x^{2} - 62\,n x^{3} - 5\,n x^{4} - 54\,n^{2} - 105\,n^{2} x - 60\,n^{2} x^{2} - 5\,n^{2} x^{3} - 18\,n^{3} - 15\,n^{3} x\right)\,P_{n-1}(x) + \left(216 + 780\,x + 1086\,x^{2} + 719\,x^{3} + 222\,x^{4} + 26\,x^{5} + x^{6} + 180\,n + 504\,n x + 495\,n x^{2} + 195\,n x^{3} + 25\,n x^{4} + n x^{5} + 36\,n^{2} + 72\,n^{2} x + 39\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17310
22218343
37696333208
4244420234483949
5772169213204446334294
6239263846456309672678319135
773122304028728180726084108093384412

All 15 rows

Technical details
Node id
n-b2243e58eef51d34 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3a5862813a018c19

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:     7,      3;
  2:    22,     18,      3;
  3:    76,     96,     33,      3;
  4:   244,    420,    234,     48,      3;
  5:   772,   1692,   1320,    444,     63,     3;
  6:  2392,   6384,   6456,   3096,    726,    78,     3;
  7:  7312,  23040,  28728,  18072,   6084,  1080,    93,    3;
  8: 22096,  80352, 119376,  93456,  41580, 10620,  1506,  108,   3;
  9: 66160, 272880, 471168, 442656, 246636, 83556, 17040, 2004, 123, 3;
%F Formula
The row polynomials satisfy -(12*n-(3*n^3+12*n^2-21*n-24)*x-(12*n^2+2*n-36)*x^2-(n^2+10*n-13)*x^3-(n-1)*x^4)*P(n,x) = -(18*n^3+54*n^2+12*n+(15*n^3+105*n^2+102*n+24)*x+(60*n^2+148*n+36)*x^2+(5*n^2+62*n+13)*x^3+(5*n+1)*x^4)*P(n-1,x) + (36*n^2+180*n+216+(72*n^2+504*n+780)*x+(39*n^2+495*n+1086)*x^2+(3*n^2+195*n+719)*x^3+(25*n+222)*x^4+(n+26)*x^5+x^6)*d/dx 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^3 + 54*n^2 + 12*n + (15*n^3 + 105*n^2 + 102*n + 24)*x + (60*n^2 + 148*n + 36)*x^2 + (5*n^2 + 62*n + 13)*x^3 + (5*n + 1)*x^4)*P[n - 1, x] + (36*n^2 + 180*n + 216 + (72*n^2 + 504*n + 780)*x + (39*n^2 + 495*n + 1086)*x^2 + (3*n^2 + 195*n + 719)*x^3 + (25*n + 222)*x^4 + (n + 26)*x^5 + x^6)*D[P[n - 1, x], x])/(-12*n + (3*n^3 + 12*n^2 - 21*n - 24)*x + (12*n^2 + 2*n - 36)*x^2 + (n^2 + 10*n - 13)*x^3 + (n - 1)*x^4)]];
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^3 + 54*n^2 + 12*n + (15*n^3 + 105*n^2 + 102*n + 24)*x + (60*n^2 + 148*n + 36)*x^2 + (5*n^2 + 62*n + 13)*x^3 + (5*n + 1)*x^4)*v[i - 1] + (36*n^2 + 180*n + 216 + (72*n^2 + 504*n + 780)*x + (39*n^2 + 495*n + 1086)*x^2 + (3*n^2 + 195*n + 719)*x^3 + (25*n + 222)*x^4 + (n + 26)*x^5 + x^6)*deriv(v[i - 1], x))/(-12*n + (3*n^3 + 12*n^2 - 21*n - 24)*x + (12*n^2 + 2*n - 36)*x^2 + (n^2 + 10*n - 13)*x^3 + (n - 1)*x^4)); 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.