Real-rooted atlas

A059473 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(9\,x^{2} + 6\,x^{3} + 4\,n - n x - 15\,n x^{2} - 9\,n x^{3} + 5\,n^{2} x + 6\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(20\,x^{2} + 33\,x^{3} + 12\,x^{4} - 4\,n x - 16\,n x^{2} - 18\,n x^{3} - 6\,n x^{4}\right)\,P_{n-1}'(x) + \left(4\,x^{2} + 14\,x^{3} + 12\,x^{4} + 3\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1448
216361870
36422424080608
42561216203214243565284
51024614414016152327920158445920
640962969686016127552102368422887048399064
7163841392644894729226241008640640768219520313603468032

All 15 rows

Technical details
Node id
n-2791575c0d8044ab · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-59f9ac5c08f7a53c

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:      4,       4;
  2:     16,      36,       18;
  3:     64,     224,      240,       80;
  4:    256,    1216,     2032,     1424,      356;
  5:   1024,    6144,    14016,    15232,     7920,     1584;
  6:   4096,   29696,    86016,   127552,   102368,    42288,     7048;
  7:  16384,  139264,   489472,   922624,  1008640,   640768,   219520,    31360;
  8:  65536,  638976,  2641920,  6051840,  8407296,  7262208,  3815040,  1116288,  139536;
  9: 262144, 2883584, 13713408, 37011456, 62505984, 68557824, 48889344, 21884928, 5587776, 620864;
%F Formula
The row polynomials satisfy n*P(n,x) = (4*n+(5*n^2-n)*x+(6*n^2-15*n+9)*x^2+(3*n^2-9*n+6)*x^3)*P(n-1,x) - (4*n*x+(16*n-20)*x^2+(18*n-33)*x^3+(6*n-12)*x^4)*d/dx P(n-1,x) + (4*x^2+14*x^3+12*x^4+3*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[((4*n + (5*n^2 - n)*x + (6*n^2 - 15*n + 9)*x^2 + (3*n^2 - 9*n + 6)*x^3)*P[n - 1, x] - (4*n*x + (16*n - 20)*x^2 + (18*n - 33)*x^3 + (6*n - 12)*x^4)*D[P[n - 1, x], x] + (4*x^2 + 14*x^3 + 12*x^4 + 3*x^5)*D[P[n - 1, x], {x, 2}])/(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] = ((4*n + (5*n^2 - n)*x + (6*n^2 - 15*n + 9)*x^2 + (3*n^2 - 9*n + 6)*x^3)*v[i - 1] - (4*n*x + (16*n - 20)*x^2 + (18*n - 33)*x^3 + (6*n - 12)*x^4)*deriv(v[i - 1], x) + (4*x^2 + 14*x^3 + 12*x^4 + 3*x^5)*deriv(deriv(v[i - 1], x), x))/(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