Real-rooted atlas

A114687 → θ+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(n + n^{2}\right)\,P_{n}(x) = \left(3\,n + 2\,n x + 7\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-1 - 3\,x - 2\,x^{2} - 2\,n - 6\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1549
225361273
312926416832593
468118001680640804881
5365311820145208320216019240665
6198257585211550088480352806720448342105
71085454792488705768355204513601344001971210242900385

All 15 rows

Technical details
Node id
n-8278948ae532c997 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-06e426b054a33769

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:       5,        4;
  2:      25,       36,       12;
  3:     129,      264,      168,       32;
  4:     681,     1800,     1680,      640,       80;
  5:    3653,    11820,    14520,     8320,     2160,      192;
  6:   19825,    75852,   115500,    88480,    35280,     6720,     448;
  7:  108545,   479248,   870576,   835520,   451360,   134400,   19712,    1024;
  8:  598417,  2994192,  6318144,  7289856,  4979520,  2032128,  473088,   55296,   2304;
  9: 3317445, 18548820, 44583120, 60103680, 49694400, 25853184, 8332800, 1566720, 149760, 5120;
%F Formula
The row polynomials satisfy (n^2+n)*P(n,x) = (7*n^2+3*n+(6*n^2+2*n)*x)*P(n-1,x) - (2*n+1+(6*n+3)*x+(4*n+2)*x^2)*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[((7*n^2 + 3*n + (6*n^2 + 2*n)*x)*P[n - 1, x] - (2*n + 1 + (6*n + 3)*x + (4*n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + n)]];
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] = ((7*n^2 + 3*n + (6*n^2 + 2*n)*x)*v[i - 1] - (2*n + 1 + (6*n + 3)*x + (4*n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + n)); 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. [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