Real-rooted atlas

A078812 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(20 + 6\,x - 2\,x^{2} - 20\,n - 6\,n x + 6\,n x^{2} - 6\,n^{2} x^{2} + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-30\,n - 19\,n x - 3\,n x^{2} - 5\,n^{2} x - 6\,n^{2} x^{2} - n^{2} x^{3} + 3\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 45\,x - 23\,x^{2} - 3\,x^{3} - 5\,n x^{2} - 6\,n x^{3} - n x^{4} + 5\,n^{2} x^{2} + 6\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2325
3812323
42150274102
555180153485441
61446007023447561871
737719082853192065010877823
8987587810719922042751098147832332

All 15 rows

Technical details
Node id
n-2e61a644950f6fb1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-754d74333aa59564

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    3,     2;
   3:    8,    12,      3;
   4:   21,    50,     27,      4;
   5:   55,   180,    153,     48,      5;
   6:  144,   600,    702,    344,     75,     6;
   7:  377,  1908,   2853,   1920,    650,   108,     7;
   8:  987,  5878,  10719,   9220,   4275,  1098,   147,    8;
   9: 2584, 17700,  38121,  40032,  23700,  8316,  1715,  192,   9;
  10: 6765, 52390, 130194, 161704, 116865, 52290, 14700, 2528, 243, 10;
%F Formula
The row polynomials satisfy -(20*n-20+(6*n-6)*x-(2*n^3-6*n^2+6*n-2)*x^2)*P(n,x) = -(30*n+(5*n^2+19*n)*x-(3*n^3-6*n^2-3*n)*x^2-(n^3-n^2)*x^3)*P(n-1,x) - (25+45*x-(5*n^2-5*n-23)*x^2-(6*n^2-6*n-3)*x^3-(n^2-n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(30*n + (5*n^2 + 19*n)*x - (3*n^3 - 6*n^2 - 3*n)*x^2 - (n^3 - n^2)*x^3)*P[n - 1, x] - (25 + 45*x - (5*n^2 - 5*n - 23)*x^2 - (6*n^2 - 6*n - 3)*x^3 - (n^2 - n)*x^4)*D[P[n - 1, x], x])/(-20*n + 20 - (6*n - 6)*x + (2*n^3 - 6*n^2 + 6*n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(30*n + (5*n^2 + 19*n)*x - (3*n^3 - 6*n^2 - 3*n)*x^2 - (n^3 - n^2)*x^3)*v[i - 1] - (25 + 45*x - (5*n^2 - 5*n - 23)*x^2 - (6*n^2 - 6*n - 3)*x^3 - (n^2 - n)*x^4)*deriv(v[i - 1], x))/(-20*n + 20 - (6*n - 6)*x + (2*n^3 - 6*n^2 + 6*n - 2)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023