Real-rooted atlas

A121574 → reverse → D

nonnegative coefficients

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

Recurrence

\(\left(-4 - 8\,x + 8\,x^{2} + 7\,n + 2\,n x - 8\,n x^{2} - 3\,n^{2} + 6\,n^{2} x\right)\,P_{n}(x) = \left(-1 - 10\,x - 44\,x^{2} - 32\,x^{3} + 4\,n + 50\,n x + 52\,n x^{2} - 3\,n^{2} - 12\,n^{2} x\right)\,P_{n-1}(x) + \left(-x - 12\,x^{2} - 36\,x^{3} - 16\,x^{4} + 3\,n x + 24\,n x^{2} + 12\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
25813
38322464
4117413264281
5141344024481601158
617212915169613603844584
720308175246726160384089617648

All 15 rows

Technical details
Node id
n-6a03aaa8bbf02e21 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-e34dd69700f081ab

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:  5,   8;
  3:  8,  32,   24;
  4: 11,  74,  132,    64;
  5: 14, 134,  402,   448,   160;
  6: 17, 212,  915,  1696,  1360,   384;
  7: 20, 308, 1752,  4672,  6160,  3840,   896;
  8: 23, 422, 2994, 10564, 19960, 20256, 10304,  2048;
  9: 26, 554, 4722, 20884, 52210, 75552, 62048, 26624, 4608;
%F Formula
The row polynomials satisfy -(3*n^2-7*n+4-(6*n^2+2*n-8)*x+(8*n-8)*x^2)*P(n,x) = -(3*n^2-4*n+1+(12*n^2-50*n+10)*x-(52*n-44)*x^2+32*x^3)*P(n-1,x) + ((3*n-1)*x+(24*n-12)*x^2+(12*n-36)*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[(-(3*n^2 - 4*n + 1 + (12*n^2 - 50*n + 10)*x - (52*n - 44)*x^2 + 32*x^3)*P[n - 1, x] + ((3*n - 1)*x + (24*n - 12)*x^2 + (12*n - 36)*x^3 - 16*x^4)*D[P[n - 1, x], x])/(-3*n^2 + 7*n - 4 + (6*n^2 + 2*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] = (-(3*n^2 - 4*n + 1 + (12*n^2 - 50*n + 10)*x - (52*n - 44)*x^2 + 32*x^3)*v[i - 1] + ((3*n - 1)*x + (24*n - 12)*x^2 + (12*n - 36)*x^3 - 16*x^4)*deriv(v[i - 1], x))/(-3*n^2 + 7*n - 4 + (6*n^2 + 2*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.