Real-rooted atlas

delannoy → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3\,n x + 6\,n x^{2} + 2\,n x^{3} + 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{3} x^{3}\right)\,P_{n}(x) = \left(4 + 21\,x + 38\,x^{2} + 28\,x^{3} + 8\,x^{4} - 4\,n - 9\,n x + 7\,n x^{2} + 25\,n x^{3} + 12\,n x^{4} - 6\,n^{2} x - 13\,n^{2} x^{2} - 2\,n^{2} x^{3} + 4\,n^{2} x^{4} - 2\,n^{3} x^{2} - 3\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(2 + 14\,x + 37\,x^{2} + 45\,x^{3} + 24\,x^{4} + 4\,x^{5} + 4\,n x + 22\,n x^{2} + 42\,n x^{3} + 31\,n x^{4} + 6\,n x^{5} + 2\,n^{2} x^{2} + 8\,n^{2} x^{3} + 9\,n^{2} x^{4} + 2\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
25201237
312729632212
429232480352801173
5707001968227211201926322
616920287212114248880326444833425
740857122457649600531203072089601024174120

All 15 rows

Technical details
Node id
n-1432bc31355d47d2 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8b913131e1582cd7

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,     4;
  2:    5,    20,     12;
  3:   12,    72,     96,     32;
  4:   29,   232,    480,    352,      80;
  5:   70,   700,   1968,   2272,    1120,     192;
  6:  169,  2028,   7212,  11424,    8880,    3264,    448;
  7:  408,  5712,  24576,  49600,   53120,   30720,   8960,   1024;
  8:  985, 15760,  79584, 195392,  267680,  213504,  97664,  23552,  2304;
  9: 2378, 42804, 248160, 718144, 1200960, 1227648, 774144, 291840, 59904, 5120;
%F Formula
The row polynomials satisfy (3*n*x+(3*n^2+6*n)*x^2+(n^3+3*n^2+2*n)*x^3)*P(n,x) = -(4*n-4+(6*n^2+9*n-21)*x+(2*n^3+13*n^2-7*n-38)*x^2+(3*n^3+2*n^2-25*n-28)*x^3-(4*n^2+12*n+8)*x^4)*P(n-1,x) + (2+(4*n+14)*x+(2*n^2+22*n+37)*x^2+(8*n^2+42*n+45)*x^3+(9*n^2+31*n+24)*x^4+(2*n^2+6*n+4)*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[(-(4*n - 4 + (6*n^2 + 9*n - 21)*x + (2*n^3 + 13*n^2 - 7*n - 38)*x^2 + (3*n^3 + 2*n^2 - 25*n - 28)*x^3 - (4*n^2 + 12*n + 8)*x^4)*P[n - 1, x] + (2 + (4*n + 14)*x + (2*n^2 + 22*n + 37)*x^2 + (8*n^2 + 42*n + 45)*x^3 + (9*n^2 + 31*n + 24)*x^4 + (2*n^2 + 6*n + 4)*x^5)*D[P[n - 1, x], x])/(3*n*x + (3*n^2 + 6*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] = (-(4*n - 4 + (6*n^2 + 9*n - 21)*x + (2*n^3 + 13*n^2 - 7*n - 38)*x^2 + (3*n^3 + 2*n^2 - 25*n - 28)*x^3 - (4*n^2 + 12*n + 8)*x^4)*v[i - 1] + (2 + (4*n + 14)*x + (2*n^2 + 22*n + 37)*x^2 + (8*n^2 + 42*n + 45)*x^3 + (9*n^2 + 31*n + 24)*x^4 + (2*n^2 + 6*n + 4)*x^5)*deriv(v[i - 1], x))/(3*n*x + (3*n^2 + 6*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.