Real-rooted atlas

delannoy → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(12\,n x + 12\,n x^{2} + 2\,n x^{3} + 6\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{3} x^{3}\right)\,P_{n}(x) = \left(32 + 84\,x + 76\,x^{2} + 28\,x^{3} + 4\,x^{4} - 32\,n - 36\,n x + 14\,n x^{2} + 25\,n x^{3} + 6\,n x^{4} - 24\,n^{2} x - 26\,n^{2} x^{2} - 2\,n^{2} x^{3} + 2\,n^{2} x^{4} - 4\,n^{3} x^{2} - 3\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(32 + 112\,x + 148\,x^{2} + 90\,x^{3} + 24\,x^{4} + 2\,x^{5} + 32\,n x + 88\,n x^{2} + 84\,n x^{3} + 31\,n x^{4} + 3\,n x^{5} + 8\,n^{2} x^{2} + 16\,n^{2} x^{3} + 9\,n^{2} x^{4} + n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2510318
3123624476
429116120445314
5703504922847061272
616910141803142855510275078
74082856614462003320960140820036

All 15 rows

Technical details
Node id
n-b6ef9a8a42517069 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-06a4c747287235b2

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,     2;
  2:    5,    10,     3;
  3:   12,    36,    24,     4;
  4:   29,   116,   120,    44,     5;
  5:   70,   350,   492,   284,    70,     6;
  6:  169,  1014,  1803,  1428,   555,   102,     7;
  7:  408,  2856,  6144,  6200,  3320,   960,   140,    8;
  8:  985,  7880, 19896, 24424, 16730,  6672,  1526,  184,   9;
  9: 2378, 21402, 62040, 89768, 75060, 38364, 12096, 2280, 234, 10;
%F Formula
The row polynomials satisfy (12*n*x+(6*n^2+12*n)*x^2+(n^3+3*n^2+2*n)*x^3)*P(n,x) = -(32*n-32+(24*n^2+36*n-84)*x+(4*n^3+26*n^2-14*n-76)*x^2+(3*n^3+2*n^2-25*n-28)*x^3-(2*n^2+6*n+4)*x^4)*P(n-1,x) + (32+(32*n+112)*x+(8*n^2+88*n+148)*x^2+(16*n^2+84*n+90)*x^3+(9*n^2+31*n+24)*x^4+(n^2+3*n+2)*x^5)*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[(-(32*n - 32 + (24*n^2 + 36*n - 84)*x + (4*n^3 + 26*n^2 - 14*n - 76)*x^2 + (3*n^3 + 2*n^2 - 25*n - 28)*x^3 - (2*n^2 + 6*n + 4)*x^4)*P[n - 1, x] + (32 + (32*n + 112)*x + (8*n^2 + 88*n + 148)*x^2 + (16*n^2 + 84*n + 90)*x^3 + (9*n^2 + 31*n + 24)*x^4 + (n^2 + 3*n + 2)*x^5)*D[P[n - 1, x], x])/(12*n*x + (6*n^2 + 12*n)*x^2 + (n^3 + 3*n^2 + 2*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] = (-(32*n - 32 + (24*n^2 + 36*n - 84)*x + (4*n^3 + 26*n^2 - 14*n - 76)*x^2 + (3*n^3 + 2*n^2 - 25*n - 28)*x^3 - (2*n^2 + 6*n + 4)*x^4)*v[i - 1] + (32 + (32*n + 112)*x + (8*n^2 + 88*n + 148)*x^2 + (16*n^2 + 84*n + 90)*x^3 + (9*n^2 + 31*n + 24)*x^4 + (n^2 + 3*n + 2)*x^5)*deriv(v[i - 1], x))/(12*n*x + (6*n^2 + 12*n)*x^2 + (n^3 + 3*n^2 + 2*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.