Real-rooted atlas

delannoy → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(9\,n\right)\,P_{n}(x) = \left(99 + 54\,x - 72\,n - 27\,n x + 18\,n^{2}\right)\,P_{n-1}(x) + \left(375 + 423\,x + 108\,x^{2} - 123\,n - 126\,n x - 27\,n x^{2}\right)\,P_{n-1}'(x) + \left(164 + 291\,x + 162\,x^{2} + 27\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1538
22933971
316527915327624
494121091737621815489
5536514979164618883234924348280
630589102249140517101493405818505729424663
7174405679023112011310139315434291722872988921873735264

All 15 rows

Technical details
Node id
n-a6c565793e739bce · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-32e041d93e8340e1

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:       5,        3;
  2:      29,       33,        9;
  3:     165,      279,      153,       27;
  4:     941,     2109,     1737,      621,       81;
  5:    5365,    14979,    16461,     8883,     2349,      243;
  6:   30589,   102249,   140517,   101493,    40581,     8505,      729;
  7:  174405,   679023,  1120113,  1013931,   543429,   172287,    29889,    2187;
  8:  994381,  4419093,  8506449,  9257517,  6225741,  2647485,   694737,  102789,   6561;
  9: 5669525, 28317915, 62307045, 79222995, 64116765, 34234083, 12052557, 2696571, 347733, 19683;
%F Formula
The row polynomials satisfy 9*n*P(n,x) = (18*n^2-72*n+99-(27*n-54)*x)*P(n-1,x) - (123*n-375+(126*n-423)*x+(27*n-108)*x^2)*d/dx P(n-1,x) + (164+291*x+162*x^2+27*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[((18*n^2 - 72*n + 99 - (27*n - 54)*x)*P[n - 1, x] - (123*n - 375 + (126*n - 423)*x + (27*n - 108)*x^2)*D[P[n - 1, x], x] + (164 + 291*x + 162*x^2 + 27*x^3)*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] = ((18*n^2 - 72*n + 99 - (27*n - 54)*x)*v[i - 1] - (123*n - 375 + (126*n - 423)*x + (27*n - 108)*x^2)*deriv(v[i - 1], x) + (164 + 291*x + 162*x^2 + 27*x^3)*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

Candidate equalities ?

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