Real-rooted atlas

delannoy → basis A013611 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(26 + 16\,x - 18\,n - 8\,n x + 4\,n^{2}\right)\,P_{n-1}(x) + \left(88 + 110\,x + 32\,x^{2} - 28\,n - 32\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x) + \left(35 + 68\,x + 42\,x^{2} + 8\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
2415216109
3276520320641180
4186146484288172825612781
51254639036480002912087041024138430
6845813150844841763925121768964198440961499329
757021624741924561536462937627914249994241966081638416239160

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(5 + 4\,x\right)\,P_{n-2}(x) + \left(6 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 4\,x\)
Technical details
Node id
n-0bd2f50af4f9b825 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-8643ed74bedf12c7)
Fingerprint
f6-c7d411e7f35c62e3

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:        6,         4;
  2:       41,        52,        16;
  3:      276,       520,       320,        64;
  4:     1861,      4648,      4288,      1728,       256;
  5:    12546,     39036,     48000,     29120,      8704,      1024;
  6:    84581,    315084,    484176,    392512,    176896,     41984,      4096;
  7:   570216,   2474192,   4561536,   4629376,   2791424,    999424,    196608,    16384;
  8:  3844201,  19039760,  40947200,  49921664,  37720576,  18079744,   5365760,   901120,   65536;
  9: 25916286, 144267188, 354546688, 504711808, 458484736, 275523584, 109494272, 27738112, 4063232, 262144;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (4*n^2-18*n+26-(8*n-16)*x)*P(n-1,x) - (28*n-88+(32*n-110)*x+(8*n-32)*x^2)*d/dx P(n-1,x) + (35+68*x+42*x^2+8*x^3)*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^2 - 18*n + 26 - (8*n - 16)*x)*P[n - 1, x] - (28*n - 88 + (32*n - 110)*x + (8*n - 32)*x^2)*D[P[n - 1, x], x] + (35 + 68*x + 42*x^2 + 8*x^3)*D[P[n - 1, x], {x, 2}])/(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^2 - 18*n + 26 - (8*n - 16)*x)*v[i - 1] - (28*n - 88 + (32*n - 110)*x + (8*n - 32)*x^2)*deriv(v[i - 1], x) + (35 + 68*x + 42*x^2 + 8*x^3)*deriv(deriv(v[i - 1], x), x))/(2*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