Real-rooted atlas

A108838 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(3 + 6\,x + 3\,x^{2} + 4\,n + 8\,n x + 4\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(6 + 15\,x + 6\,x^{2} - 9\,x^{3} - 6\,x^{4} + 10\,n + 36\,n x + 44\,n x^{2} + 20\,n x^{3} + 2\,n x^{4} + 4\,n^{2} + 17\,n^{2} x + 26\,n^{2} x^{2} + 17\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(-3\,x - 15\,x^{2} - 27\,x^{3} - 21\,x^{4} - 6\,x^{5} - 2\,n x - 10\,n x^{2} - 18\,n x^{3} - 14\,n x^{4} - 4\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
15712
214402882
342182259121604
4132768165615685504674
542931359075130059225259137460
61430125844576088000943805352012536308210
7486250050219219529529761761652575308231619212588148

All 15 rows

Technical details
Node id
n-df8981bac1850ed3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-154b8c3d0164f5d5

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:     2;
  1:     5,      7;
  2:    14,     40,      28;
  3:    42,    182,     259,      121;
  4:   132,    768,    1656,     1568,      550;
  5:   429,   3135,    9075,    13005,     9225,     2591;
  6:  1430,  12584,   45760,    88000,    94380,    53520,    12536;
  7:  4862,  50050,  219219,   529529,   761761,   652575,   308231,    61921;
  8: 16796, 198016, 1014832,  2952768,  5334420,  6126848,  4368728,  1768096,   310954;
  9: 58786, 781014, 4585308, 15612324, 33972120, 48989304, 46814586, 28585830, 10120461, 1582791;
%F Formula
The row polynomials satisfy (n^2+4*n+3+(2*n^2+8*n+6)*x+(n^2+4*n+3)*x^2)*P(n,x) = (4*n^2+10*n+6+(17*n^2+36*n+15)*x+(26*n^2+44*n+6)*x^2+(17*n^2+20*n-9)*x^3+(4*n^2+2*n-6)*x^4)*P(n-1,x) - ((2*n+3)*x+(10*n+15)*x^2+(18*n+27)*x^3+(14*n+21)*x^4+(4*n+6)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.
%t Mathematica
P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 10*n + 6 + (17*n^2 + 36*n + 15)*x + (26*n^2 + 44*n + 6)*x^2 + (17*n^2 + 20*n - 9)*x^3 + (4*n^2 + 2*n - 6)*x^4)*P[n - 1, x] - ((2*n + 3)*x + (10*n + 15)*x^2 + (18*n + 27)*x^3 + (14*n + 21)*x^4 + (4*n + 6)*x^5)*D[P[n - 1, x], x])/(n^2 + 4*n + 3 + (2*n^2 + 8*n + 6)*x + (n^2 + 4*n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 10*n + 6 + (17*n^2 + 36*n + 15)*x + (26*n^2 + 44*n + 6)*x^2 + (17*n^2 + 20*n - 9)*x^3 + (4*n^2 + 2*n - 6)*x^4)*v[i - 1] - ((2*n + 3)*x + (10*n + 15)*x^2 + (18*n + 27)*x^3 + (14*n + 21)*x^4 + (4*n + 6)*x^5)*deriv(v[i - 1], x))/(n^2 + 4*n + 3 + (2*n^2 + 8*n + 6)*x + (n^2 + 4*n + 3)*x^2)); 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. [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