Real-rooted atlas

A081577 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3\,n x + 6\,n x^{2} + 2\,n x^{3} + 3\,n^{2} x^{2} + 3\,n^{2} x^{3} + n^{3} x^{3}\right)\,P_{n}(x) = \left(6 + 30\,x + 50\,x^{2} + 32\,x^{3} + 8\,x^{4} - 6\,n - 15\,n x + 4\,n x^{2} + 27\,n x^{3} + 12\,n x^{4} - 9\,n^{2} x - 21\,n^{2} x^{2} - 6\,n^{2} x^{3} + 4\,n^{2} x^{4} - 3\,n^{3} x^{2} - 5\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(3 + 21\,x + 55\,x^{2} + 65\,x^{3} + 32\,x^{4} + 4\,x^{5} + 6\,n x + 33\,n x^{2} + 62\,n x^{3} + 43\,n x^{4} + 6\,n x^{5} + 3\,n^{2} x^{2} + 12\,n^{2} x^{3} + 13\,n^{2} x^{4} + 2\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
26241242
3169612032264
444352696448801620
512012003264345614401929672
63283936136322022413920422444856712
789612544528001011209856049152116481024327744

All 15 rows

Technical details
Node id
n-260b0485ef5c3d42 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-ed4ddcf1e7198dc9

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:    6,     24,     12;
  3:   16,     96,    120,      32;
  4:   44,    352,    696,     448,      80;
  5:  120,   1200,   3264,    3456,    1440,     192;
  6:  328,   3936,  13632,   20224,   13920,    4224,     448;
  7:  896,  12544,  52800,  101120,   98560,   49152,   11648,   1024;
  8: 2448,  39168, 194112,  456192,  578880,  410112,  158592,  30720,  2304;
  9: 6688, 120384, 686592, 1913856, 3000960, 2780928, 1526784, 479232, 78336, 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) = -(6*n-6+(9*n^2+15*n-30)*x+(3*n^3+21*n^2-4*n-50)*x^2+(5*n^3+6*n^2-27*n-32)*x^3-(4*n^2+12*n+8)*x^4)*P(n-1,x) + (3+(6*n+21)*x+(3*n^2+33*n+55)*x^2+(12*n^2+62*n+65)*x^3+(13*n^2+43*n+32)*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[(-(6*n - 6 + (9*n^2 + 15*n - 30)*x + (3*n^3 + 21*n^2 - 4*n - 50)*x^2 + (5*n^3 + 6*n^2 - 27*n - 32)*x^3 - (4*n^2 + 12*n + 8)*x^4)*P[n - 1, x] + (3 + (6*n + 21)*x + (3*n^2 + 33*n + 55)*x^2 + (12*n^2 + 62*n + 65)*x^3 + (13*n^2 + 43*n + 32)*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] = (-(6*n - 6 + (9*n^2 + 15*n - 30)*x + (3*n^3 + 21*n^2 - 4*n - 50)*x^2 + (5*n^3 + 6*n^2 - 27*n - 32)*x^3 - (4*n^2 + 12*n + 8)*x^4)*v[i - 1] + (3 + (6*n + 21)*x + (3*n^2 + 33*n + 55)*x^2 + (12*n^2 + 62*n + 65)*x^3 + (13*n^2 + 43*n + 32)*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

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.