Real-rooted atlas

delannoy → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(12\,x^{2} + 16\,x^{3} + n - n x - 20\,n x^{2} - 24\,n x^{3} + 3\,n^{2} x + 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(13\,x^{2} + 44\,x^{3} + 32\,x^{4} - n x - 10\,n x^{2} - 24\,n x^{3} - 16\,n x^{4}\right)\,P_{n-1}'(x) + \left(x^{2} + 9\,x^{3} + 16\,x^{4} + 8\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
215511
318181239
4111405829139
51147116417570495
61171113576015071691763
71201606641550204814284086279
81232181112334661066632394098522363
91262851728639415012224422068010701237879647
1012936125391118032258619187830262633287055741283667
111324463572182716274014784023959226235918535276230138601010295
12135540485428315113001316758629448881763851005538272200766334613598219
13138643641242041191474623771147354425280913116482268799315388925250838078212815247
1414175582736025930872511480353150845642138596838751065372583041554341905136517519502545642179
Technical details
Node id
n-a5aa3683b1f599f4 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f8b12148df73b33e

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: 1,  2;
  2: 1,  5,   5;
  3: 1,  8,  18,   12;
  4: 1, 11,  40,   58,   29;
  5: 1, 14,  71,  164,  175,    70;
  6: 1, 17, 111,  357,  601,   507,   169;
  7: 1, 20, 160,  664, 1550,  2048,  1428,   408;
  8: 1, 23, 218, 1112, 3346,  6106,  6632,  3940,   985;
  9: 1, 26, 285, 1728, 6394, 15012, 22442, 20680, 10701, 2378;
%F Formula
The row polynomials satisfy n*P(n,x) = (n+(3*n^2-n)*x+(8*n^2-20*n+12)*x^2+(8*n^2-24*n+16)*x^3)*P(n-1,x) - (n*x+(10*n-13)*x^2+(24*n-44)*x^3+(16*n-32)*x^4)*d/dx P(n-1,x) + (x^2+9*x^3+16*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[((n + (3*n^2 - n)*x + (8*n^2 - 20*n + 12)*x^2 + (8*n^2 - 24*n + 16)*x^3)*P[n - 1, x] - (n*x + (10*n - 13)*x^2 + (24*n - 44)*x^3 + (16*n - 32)*x^4)*D[P[n - 1, x], x] + (x^2 + 9*x^3 + 16*x^4 + 8*x^5)*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] = ((n + (3*n^2 - n)*x + (8*n^2 - 20*n + 12)*x^2 + (8*n^2 - 24*n + 16)*x^3)*v[i - 1] - (n*x + (10*n - 13)*x^2 + (24*n - 44)*x^3 + (16*n - 32)*x^4)*deriv(v[i - 1], x) + (x^2 + 9*x^3 + 16*x^4 + 8*x^5)*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

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