Real-rooted atlas

delannoy → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(9\,n + 27\,n x + 27\,n x^{2} + 9\,n x^{3}\right)\,P_{n}(x) = \left(121 + 933\,x + 2850\,x^{2} + 4490\,x^{3} + 3885\,x^{4} + 1761\,x^{5} + 328\,x^{6} - 153\,n - 1194\,n x - 3765\,n x^{2} - 6153\,n x^{3} - 5514\,n x^{4} - 2577\,n x^{5} - 492\,n x^{6} + 50\,n^{2} + 360\,n^{2} x + 1104\,n^{2} x^{2} + 1816\,n^{2} x^{3} + 1674\,n^{2} x^{4} + 816\,n^{2} x^{5} + 164\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(92 + 983\,x + 4191\,x^{2} + 9430\,x^{3} + 12250\,x^{4} + 9267\,x^{5} + 3803\,x^{6} + 656\,x^{7} - 40\,n - 424\,n x - 1821\,n x^{2} - 4172\,n x^{3} - 5558\,n x^{4} - 4332\,n x^{5} - 1837\,n x^{6} - 328\,n x^{7}\right)\,P_{n-1}'(x) + \left(8 + 112\,x + 641\,x^{2} + 1969\,x^{3} + 3590\,x^{4} + 4018\,x^{5} + 2717\,x^{6} + 1021\,x^{7} + 164\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1257
25252959
31290216165483
429290105616559413971
57087542761019511846536532627
616925351554949865881078128530589268099
74087140527042125505054647084805418121744052202963

All 15 rows

Technical details
Node id
n-ea41f0a7e484c4eb · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-830711f8ecdf3b7b

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:    2,     5;
  2:    5,    25,     29;
  3:   12,    90,    216,     165;
  4:   29,   290,   1056,    1655,      941;
  5:   70,   875,   4276,   10195,    11846,     5365;
  6:  169,  2535,  15549,   49865,    88107,    81285,    30589;
  7:  408,  7140,  52704,  212550,   505464,   708480,   541812,   174405;
  8:  985, 19700, 170008,  826370,  2473306,  4665560,  5415466,  3535955,   994381;
  9: 2378, 53505, 528552, 3007410, 10857492, 25783650, 40264800, 39866625, 22707810, 5669525;
%F Formula
The row polynomials satisfy (9*n+27*n*x+27*n*x^2+9*n*x^3)*P(n,x) = (50*n^2-153*n+121+(360*n^2-1194*n+933)*x+(1104*n^2-3765*n+2850)*x^2+(1816*n^2-6153*n+4490)*x^3+(1674*n^2-5514*n+3885)*x^4+(816*n^2-2577*n+1761)*x^5+(164*n^2-492*n+328)*x^6)*P(n-1,x) - (40*n-92+(424*n-983)*x+(1821*n-4191)*x^2+(4172*n-9430)*x^3+(5558*n-12250)*x^4+(4332*n-9267)*x^5+(1837*n-3803)*x^6+(328*n-656)*x^7)*d/dx P(n-1,x) + (8+112*x+641*x^2+1969*x^3+3590*x^4+4018*x^5+2717*x^6+1021*x^7+164*x^8)*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[((50*n^2 - 153*n + 121 + (360*n^2 - 1194*n + 933)*x + (1104*n^2 - 3765*n + 2850)*x^2 + (1816*n^2 - 6153*n + 4490)*x^3 + (1674*n^2 - 5514*n + 3885)*x^4 + (816*n^2 - 2577*n + 1761)*x^5 + (164*n^2 - 492*n + 328)*x^6)*P[n - 1, x] - (40*n - 92 + (424*n - 983)*x + (1821*n - 4191)*x^2 + (4172*n - 9430)*x^3 + (5558*n - 12250)*x^4 + (4332*n - 9267)*x^5 + (1837*n - 3803)*x^6 + (328*n - 656)*x^7)*D[P[n - 1, x], x] + (8 + 112*x + 641*x^2 + 1969*x^3 + 3590*x^4 + 4018*x^5 + 2717*x^6 + 1021*x^7 + 164*x^8)*D[P[n - 1, x], {x, 2}])/(9*n + 27*n*x + 27*n*x^2 + 9*n*x^3)]];
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] = ((50*n^2 - 153*n + 121 + (360*n^2 - 1194*n + 933)*x + (1104*n^2 - 3765*n + 2850)*x^2 + (1816*n^2 - 6153*n + 4490)*x^3 + (1674*n^2 - 5514*n + 3885)*x^4 + (816*n^2 - 2577*n + 1761)*x^5 + (164*n^2 - 492*n + 328)*x^6)*v[i - 1] - (40*n - 92 + (424*n - 983)*x + (1821*n - 4191)*x^2 + (4172*n - 9430)*x^3 + (5558*n - 12250)*x^4 + (4332*n - 9267)*x^5 + (1837*n - 3803)*x^6 + (328*n - 656)*x^7)*deriv(v[i - 1], x) + (8 + 112*x + 641*x^2 + 1969*x^3 + 3590*x^4 + 4018*x^5 + 2717*x^6 + 1021*x^7 + 164*x^8)*deriv(deriv(v[i - 1], x), x))/(9*n + 27*n*x + 27*n*x^2 + 9*n*x^3)); 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