Real-rooted atlas

delannoy → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-12 + 27\,x^{2} + 12\,n - 18\,n x - 27\,n x^{2} + 18\,n^{2} x\right)\,P_{n}(x) = \left(-60 - 288\,x - 405\,x^{2} - 162\,x^{3} + 108\,n + 378\,n x + 243\,n x^{2} - 24\,n^{2} - 54\,n^{2} x\right)\,P_{n-1}(x) + \left(-16 - 120\,x - 306\,x^{2} - 297\,x^{3} - 81\,x^{4} + 16\,n + 96\,n x + 162\,n x^{2} + 54\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2151833
35414481279
41747208913242109
5525295257514536121514979
61521108182891735964206554374102249
74284368641255502151361944008748015309679023

All 15 rows

Technical details
Node id
n-fcc1ca9f11255f1e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-efa4ceed12824448

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:     3;
  2:    15,     18;
  3:    54,    144,      81;
  4:   174,    720,     891,     324;
  5:   525,   2952,    5751,    4536,    1215;
  6:  1521,  10818,   28917,   35964,   20655,    4374;
  7:  4284,  36864,  125550,  215136,  194400,   87480,   15309;
  8: 11820, 119376,  494586, 1084104, 1351080,  953532,  352107,   52488;
  9: 32103, 372240, 1817802, 4863888, 7768710, 7558272, 4363065, 1364688, 177147;
%F Formula
The row polynomials satisfy (12*n-12+(18*n^2-18*n)*x-(27*n-27)*x^2)*P(n,x) = -(24*n^2-108*n+60+(54*n^2-378*n+288)*x-(243*n-405)*x^2+162*x^3)*P(n-1,x) + (16*n-16+(96*n-120)*x+(162*n-306)*x^2+(54*n-297)*x^3-81*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(24*n^2 - 108*n + 60 + (54*n^2 - 378*n + 288)*x - (243*n - 405)*x^2 + 162*x^3)*P[n - 1, x] + (16*n - 16 + (96*n - 120)*x + (162*n - 306)*x^2 + (54*n - 297)*x^3 - 81*x^4)*D[P[n - 1, x], x])/(12*n - 12 + (18*n^2 - 18*n)*x - (27*n - 27)*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(24*n^2 - 108*n + 60 + (54*n^2 - 378*n + 288)*x - (243*n - 405)*x^2 + 162*x^3)*v[i - 1] + (16*n - 16 + (96*n - 120)*x + (162*n - 306)*x^2 + (54*n - 297)*x^3 - 81*x^4)*deriv(v[i - 1], x))/(12*n - 12 + (18*n^2 - 18*n)*x - (27*n - 27)*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.