Real-rooted atlas

delannoy → basis A013609 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(9 + 4\,x - 7\,n - 2\,n x + 2\,n^{2}\right)\,P_{n-1}(x) + \left(41 + 39\,x + 8\,x^{2} - 14\,n - 12\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x) + \left(21 + 32\,x + 15\,x^{2} + 2\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
21918441
388124568276
4409764520152161861
519004422402417843843212546
6882724598280281668054569286484581
74100813311218222413617659904154882176128570216

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(3 + 2\,x\right)\,P_{n-2}(x) + \left(4 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 2\,x\)
Technical details
Node id
n-f81bd2b0431f100d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-c210d74701eaf635)
Fingerprint
f6-b4e11046d64dfe4b

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:      4,       2;
  2:     19,      18,       4;
  3:     88,     124,      56,       8;
  4:    409,     764,     520,     152,      16;
  5:   1900,    4422,    4024,    1784,     384,      32;
  6:   8827,   24598,   28028,   16680,    5456,     928,     64;
  7:  41008,  133112,  182224,  136176,   59904,   15488,   2176,    128;
  8: 190513,  705912, 1128400, 1015248,  561696,  195456,  41728,   4992,   256;
  9: 885076, 3686026, 6738320, 7090768, 4729344, 2071488, 595328, 108160, 11264, 512;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n^2-7*n+9-(2*n-4)*x)*P(n-1,x) - (14*n-41+(12*n-39)*x+(2*n-8)*x^2)*d/dx P(n-1,x) + (21+32*x+15*x^2+2*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[((2*n^2 - 7*n + 9 - (2*n - 4)*x)*P[n - 1, x] - (14*n - 41 + (12*n - 39)*x + (2*n - 8)*x^2)*D[P[n - 1, x], x] + (21 + 32*x + 15*x^2 + 2*x^3)*D[P[n - 1, x], {x, 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] = ((2*n^2 - 7*n + 9 - (2*n - 4)*x)*v[i - 1] - (14*n - 41 + (12*n - 39)*x + (2*n - 8)*x^2)*deriv(v[i - 1], x) + (21 + 32*x + 15*x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), x))/(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