Real-rooted atlas

A207606 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-42 - 24\,x^{2} - 6\,x^{3} - 26\,n + 12\,n x - 18\,n x^{2} - 4\,n x^{3} + 18\,n^{2} x + 2\,n^{3} + 4\,n^{3} x\right)\,P_{n}(x) = \left(-124 - 12\,x - 46\,x^{2} - 30\,x^{3} - 4\,x^{4} - 60\,n + 34\,n x - 15\,n x^{2} - 17\,n x^{3} - 2\,n x^{4} + 4\,n^{2} + 42\,n^{2} x + 14\,n^{2} x^{2} + 3\,n^{3} + 7\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-70 - 149\,x - 157\,x^{2} - 101\,x^{3} - 25\,x^{4} - 2\,x^{5} + 5\,n + 46\,n x + 49\,n x^{2} + 8\,n x^{3} + 5\,n^{2} + 16\,n^{2} x + 13\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 24 + 15\,x + 2\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02415241
17878232181
2244341162312780
374313448972763923301
4221849444290185442047213775
56521173141858510450332059455256841
618942584437497452211215705403798632232406
7544861917182866212388681222263977482111032712943009

All 15 rows

Technical details
Node id
n-ec01f6b0f609a210 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1375c825235e5722

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:     24,      15,       2;
  1:     78,      78,      23,       2;
  2:    244,     341,     162,      31,       2;
  3:    743,    1344,     897,     276,      39,       2;
  4:   2218,    4944,    4290,    1854,     420,      47,      2;
  5:   6521,   17314,   18585,   10450,    3320,     594,     55,      2;
  6:  18942,   58443,   74974,   52211,   21570,    5403,    798,     63,     2;
  7:  54486,  191718,  286621,  238868,  122226,   39774,   8211,   1032,    71,    2;
  8: 155462,  614737, 1050696, 1021858,  625982,  252189,  67564,  11852,  1296,   79,  2;
  9: 440557, 1934760, 3724170, 4146312, 2965815, 1430324, 474110, 107820, 16434, 1590, 87, 2;
%F Formula
The row polynomials satisfy (2*n^3-26*n-42+(4*n^3+18*n^2+12*n)*x-(18*n+24)*x^2-(4*n+6)*x^3)*P(n,x) = (3*n^3+4*n^2-60*n-124+(7*n^3+42*n^2+34*n-12)*x+(2*n^3+14*n^2-15*n-46)*x^2-(17*n+30)*x^3-(2*n+4)*x^4)*P(n-1,x) + (5*n^2+5*n-70+(16*n^2+46*n-149)*x+(13*n^2+49*n-157)*x^2+(2*n^2+8*n-101)*x^3-25*x^4-2*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 24 + 15*x + 2*x^2.
%t Mathematica
P[0, x_] := 24 + 15*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^3 + 4*n^2 - 60*n - 124 + (7*n^3 + 42*n^2 + 34*n - 12)*x + (2*n^3 + 14*n^2 - 15*n - 46)*x^2 - (17*n + 30)*x^3 - (2*n + 4)*x^4)*P[n - 1, x] + (5*n^2 + 5*n - 70 + (16*n^2 + 46*n - 149)*x + (13*n^2 + 49*n - 157)*x^2 + (2*n^2 + 8*n - 101)*x^3 - 25*x^4 - 2*x^5)*D[P[n - 1, x], x])/(2*n^3 - 26*n - 42 + (4*n^3 + 18*n^2 + 12*n)*x - (18*n + 24)*x^2 - (4*n + 6)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 24 + 15*x + 2*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((3*n^3 + 4*n^2 - 60*n - 124 + (7*n^3 + 42*n^2 + 34*n - 12)*x + (2*n^3 + 14*n^2 - 15*n - 46)*x^2 - (17*n + 30)*x^3 - (2*n + 4)*x^4)*v[i - 1] + (5*n^2 + 5*n - 70 + (16*n^2 + 46*n - 149)*x + (13*n^2 + 49*n - 157)*x^2 + (2*n^2 + 8*n - 101)*x^3 - 25*x^4 - 2*x^5)*deriv(v[i - 1], x))/(2*n^3 - 26*n - 42 + (4*n^3 + 18*n^2 + 12*n)*x - (18*n + 24)*x^2 - (4*n + 6)*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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023