Real-rooted atlas

A207606 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-6 + 6\,x - 4\,n + 10\,n x + 2\,n^{2} + 4\,n^{2} x\right)\,P_{n}(x) = \left(-33 + 21\,x + 12\,x^{2} - 2\,n + 28\,n x + 10\,n x^{2} + 3\,n^{2} + 7\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-5 + 14\,x + 23\,x^{2} + 4\,x^{3} + 5\,n + 16\,n x + 13\,n x^{2} + 2\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 11 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
011415
14434684
2155186698418
3510834480116101950
4160833362622980175128733
549241239012390632017402461438024
61476143682531333450012945281432916162180
7435401481782125321680248022023730425642418680922

All 15 rows

Technical details
Node id
n-e5e984bb93520d4a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-df99637c8a69efb1

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     11,       4;
  1:     44,      34,       6;
  2:    155,     186,      69,       8;
  3:    510,     834,     480,     116,      10;
  4:   1608,    3336,    2622,     980,     175,     12;
  5:   4924,   12390,   12390,    6320,    1740,    246,     14;
  6:  14761,   43682,   53133,   34500,   12945,   2814,    329,    16;
  7:  43540,  148178,  212532,  168024,   80220,  23730,   4256,   424,   18;
  8: 126805,  487932,  806730,  752948,  437745, 164640,  40124,  6120,  531,  20;
  9: 365532, 1569228, 2939556, 3166460, 2174200, 995484, 308196, 63792, 8460, 650, 22;
%F Formula
The row polynomials satisfy (2*n^2-4*n-6+(4*n^2+10*n+6)*x)*P(n,x) = (3*n^2-2*n-33+(7*n^2+28*n+21)*x+(2*n^2+10*n+12)*x^2)*P(n-1,x) + (5*n-5+(16*n+14)*x+(13*n+23)*x^2+(2*n+4)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 11 + 4*x.
%t Mathematica
P[0, x_] := 11 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 2*n - 33 + (7*n^2 + 28*n + 21)*x + (2*n^2 + 10*n + 12)*x^2)*P[n - 1, x] + (5*n - 5 + (16*n + 14)*x + (13*n + 23)*x^2 + (2*n + 4)*x^3)*D[P[n - 1, x], x])/(2*n^2 - 4*n - 6 + (4*n^2 + 10*n + 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 11 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = ((3*n^2 - 2*n - 33 + (7*n^2 + 28*n + 21)*x + (2*n^2 + 10*n + 12)*x^2)*v[i - 1] + (5*n - 5 + (16*n + 14)*x + (13*n + 23)*x^2 + (2*n + 4)*x^3)*deriv(v[i - 1], x))/(2*n^2 - 4*n - 6 + (4*n^2 + 10*n + 6)*x)); 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