Real-rooted atlas

delannoy → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + 4\,x^{2} + n - 2\,n x - 4\,n x^{2} + 2\,n^{2} x\right)\,P_{n}(x) = \left(-5 - 32\,x - 60\,x^{2} - 32\,x^{3} + 9\,n + 42\,n x + 36\,n x^{2} - 2\,n^{2} - 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-1 - 10\,x - 34\,x^{2} - 44\,x^{3} - 16\,x^{4} + n + 8\,n x + 18\,n x^{2} + 8\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
144
2203252
372256192520
42321280211210244648
570052481363214336512039036
6202819232685441136648704024576315084
75712655362976006799368192004915201146882474192

All 15 rows

Technical details
Node id
n-ad143d5196921079 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-fb879c47f0d3aaa6

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:    20,     32;
  3:    72,    256,     192;
  4:   232,   1280,    2112,     1024;
  5:   700,   5248,   13632,    14336,     5120;
  6:  2028,  19232,   68544,   113664,    87040,    24576;
  7:  5712,  65536,  297600,   679936,   819200,   491520,   114688;
  8: 15760, 212224, 1172352,  3426304,  5693440,  5357568,  2637824,   524288;
  9: 42804, 661760, 4308864, 15372288, 32737280, 42467328, 32686080, 13631488, 2359296;
%F Formula
The row polynomials satisfy (n-1+(2*n^2-2*n)*x-(4*n-4)*x^2)*P(n,x) = -(2*n^2-9*n+5+(6*n^2-42*n+32)*x-(36*n-60)*x^2+32*x^3)*P(n-1,x) + (n-1+(8*n-10)*x+(18*n-34)*x^2+(8*n-44)*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[(-(2*n^2 - 9*n + 5 + (6*n^2 - 42*n + 32)*x - (36*n - 60)*x^2 + 32*x^3)*P[n - 1, x] + (n - 1 + (8*n - 10)*x + (18*n - 34)*x^2 + (8*n - 44)*x^3 - 16*x^4)*D[P[n - 1, x], x])/(n - 1 + (2*n^2 - 2*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] = (-(2*n^2 - 9*n + 5 + (6*n^2 - 42*n + 32)*x - (36*n - 60)*x^2 + 32*x^3)*v[i - 1] + (n - 1 + (8*n - 10)*x + (18*n - 34)*x^2 + (8*n - 44)*x^3 - 16*x^4)*deriv(v[i - 1], x))/(n - 1 + (2*n^2 - 2*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

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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.