Real-rooted atlas

delannoy → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(9\,n\right)\,P_{n}(x) = \left(36\,x^{2} + 16\,x^{3} + 27\,n - 9\,n x - 60\,n x^{2} - 24\,n x^{3} + 27\,n^{2} x + 24\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(117\,x^{2} + 132\,x^{3} + 32\,x^{4} - 27\,n x - 90\,n x^{2} - 72\,n x^{3} - 16\,n x^{4}\right)\,P_{n-1}'(x) + \left(27\,x^{2} + 81\,x^{3} + 48\,x^{4} + 8\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
2915529
327725412165
48129736017429941
5243113419171476525705365
67294131899196395409152116930589
7218714580388805378441850184324284408174405

All 15 rows

Technical details
Node id
n-69ac01f25ec67ed3 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-cfb345b8c9d5b7ff

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:     3,      2;
  2:     9,     15,      5;
  3:    27,     72,     54,      12;
  4:    81,    297,    360,     174,      29;
  5:   243,   1134,   1917,    1476,     525,      70;
  6:   729,   4131,   8991,    9639,    5409,    1521,    169;
  7:  2187,  14580,  38880,   53784,   41850,   18432,   4284,    408;
  8:  6561,  50301, 158922,  270216,  271026,  164862,  59688,  11820,   985;
  9: 19683, 170586, 623295, 1259712, 1553742, 1215972, 605934, 186120, 32103, 2378;
%F Formula
The row polynomials satisfy 9*n*P(n,x) = (27*n+(27*n^2-9*n)*x+(24*n^2-60*n+36)*x^2+(8*n^2-24*n+16)*x^3)*P(n-1,x) - (27*n*x+(90*n-117)*x^2+(72*n-132)*x^3+(16*n-32)*x^4)*d/dx P(n-1,x) + (27*x^2+81*x^3+48*x^4+8*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[((27*n + (27*n^2 - 9*n)*x + (24*n^2 - 60*n + 36)*x^2 + (8*n^2 - 24*n + 16)*x^3)*P[n - 1, x] - (27*n*x + (90*n - 117)*x^2 + (72*n - 132)*x^3 + (16*n - 32)*x^4)*D[P[n - 1, x], x] + (27*x^2 + 81*x^3 + 48*x^4 + 8*x^5)*D[P[n - 1, x], {x, 2}])/(9*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] = ((27*n + (27*n^2 - 9*n)*x + (24*n^2 - 60*n + 36)*x^2 + (8*n^2 - 24*n + 16)*x^3)*v[i - 1] - (27*n*x + (90*n - 117)*x^2 + (72*n - 132)*x^3 + (16*n - 32)*x^4)*deriv(v[i - 1], x) + (27*x^2 + 81*x^3 + 48*x^4 + 8*x^5)*deriv(deriv(v[i - 1], x), x))/(9*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