Real-rooted atlas

delannoy → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 4\,n x + 3\,n x^{2} + 6\,n x^{3} + n x^{4} + 4\,n^{2} + 6\,n^{2} x + 7\,n^{2} x^{2} + n^{2} x^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(-8 + 8\,x + 50\,x^{2} + 45\,x^{3} + 12\,x^{4} + x^{5} + 36\,n + 56\,n x + 11\,n x^{2} - 11\,n x^{3} - 2\,n x^{4} + 4\,n^{2} - 14\,n^{2} x - 19\,n^{2} x^{2} - 3\,n^{2} x^{3} - 8\,n^{3} - 6\,n^{3} x\right)\,P_{n-1}(x) + \left(32 + 152\,x + 276\,x^{2} + 239\,x^{3} + 99\,x^{4} + 17\,x^{5} + x^{6} + 48\,n + 152\,n x + 174\,n x^{2} + 85\,n x^{3} + 16\,n x^{4} + n x^{5} + 16\,n^{2} + 32\,n^{2} x + 18\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
2107118
3303411176
48713873151314
52455033771271911272
6676170916728011962315078
71836552466984206146428027120036

All 15 rows

Technical details
Node id
n-412287924c07c5fe · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8df4cfbffdd1234a

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:     3,     1;
  2:    10,     7,     1;
  3:    30,    34,    11,     1;
  4:    87,   138,    73,    15,     1;
  5:   245,   503,   377,   127,    19,     1;
  6:   676,  1709,  1672,   801,   196,    23,    1;
  7:  1836,  5524,  6698,  4206,  1464,   280,   27,   1;
  8:  4925, 17204, 24950, 19490,  8906,  2420,  379,  31,  1;
  9: 13079, 52061, 88006, 82490, 46982, 16762, 3723, 493, 35, 1;
%F Formula
The row polynomials satisfy (4*n^2+4*n+(2*n^3+6*n^2+4*n)*x+(7*n^2+3*n)*x^2+(n^2+6*n)*x^3+n*x^4)*P(n,x) = -(8*n^3-4*n^2-36*n+8+(6*n^3+14*n^2-56*n-8)*x+(19*n^2-11*n-50)*x^2+(3*n^2+11*n-45)*x^3+(2*n-12)*x^4-x^5)*P(n-1,x) + (16*n^2+48*n+32+(32*n^2+152*n+152)*x+(18*n^2+174*n+276)*x^2+(2*n^2+85*n+239)*x^3+(16*n+99)*x^4+(n+17)*x^5+x^6)*d/dx 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[(-(8*n^3 - 4*n^2 - 36*n + 8 + (6*n^3 + 14*n^2 - 56*n - 8)*x + (19*n^2 - 11*n - 50)*x^2 + (3*n^2 + 11*n - 45)*x^3 + (2*n - 12)*x^4 - x^5)*P[n - 1, x] + (16*n^2 + 48*n + 32 + (32*n^2 + 152*n + 152)*x + (18*n^2 + 174*n + 276)*x^2 + (2*n^2 + 85*n + 239)*x^3 + (16*n + 99)*x^4 + (n + 17)*x^5 + x^6)*D[P[n - 1, x], x])/(4*n^2 + 4*n + (2*n^3 + 6*n^2 + 4*n)*x + (7*n^2 + 3*n)*x^2 + (n^2 + 6*n)*x^3 + n*x^4)]];
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] = (-(8*n^3 - 4*n^2 - 36*n + 8 + (6*n^3 + 14*n^2 - 56*n - 8)*x + (19*n^2 - 11*n - 50)*x^2 + (3*n^2 + 11*n - 45)*x^3 + (2*n - 12)*x^4 - x^5)*v[i - 1] + (16*n^2 + 48*n + 32 + (32*n^2 + 152*n + 152)*x + (18*n^2 + 174*n + 276)*x^2 + (2*n^2 + 85*n + 239)*x^3 + (16*n + 99)*x^4 + (n + 17)*x^5 + x^6)*deriv(v[i - 1], x))/(4*n^2 + 4*n + (2*n^3 + 6*n^2 + 4*n)*x + (7*n^2 + 3*n)*x^2 + (n^2 + 6*n)*x^3 + n*x^4)); 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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.