Real-rooted atlas

A172431 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-60\,n - 378\,n x + 216\,n x^{2} - 144\,n^{2} + 432\,n^{2} x + 216\,n^{3}\right)\,P_{n}(x) = \left(78 + 396\,x + 270\,x^{2} - 648\,x^{3} - 690\,n - 2862\,n x - 1890\,n x^{2} + 1944\,n x^{3} - 144\,n^{2} + 1080\,n^{2} x + 3888\,n^{2} x^{2} + 864\,n^{3} + 1944\,n^{3} x\right)\,P_{n-1}(x) + \left(5 - 18\,x - 387\,x^{2} - 1296\,x^{3} - 1296\,x^{4} - 150\,n - 1170\,n x - 3024\,n x^{2} - 2592\,n x^{3} - 180\,n^{2} - 972\,n^{2} x - 1296\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
19615
2388427149
31355105221081275
444523884239270040510177
514049792250022797212555145878183
6430136972123768201312160461544325103586349
71289713189854761411766601376190836892224532174964324179

All 15 rows

Technical details
Node id
n-f1db220002d3e457 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-00cda81caf61e355

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:      9,       6;
  2:     38,      84,      27;
  3:    135,     510,     522,      108;
  4:    445,    2388,    4239,     2700,      405;
  5:   1404,    9792,   25002,    27972,    12555,     1458;
  6:   4301,   36972,  123768,   201312,   160461,    54432,     5103;
  7:  12897,  131898,  547614,  1176660,  1376190,   836892,   224532,    17496;
  8:  38050,  451428, 2240109,  6002316,  9335655,  8398080,  4072194,   892296,   59049;
  9: 110835, 1496874, 8647362, 27807300, 54196695, 65218770, 47116728, 18790704, 3444525, 196830;
%F Formula
The row polynomials satisfy (216*n^3-144*n^2-60*n+(432*n^2-378*n)*x+216*n*x^2)*P(n,x) = (864*n^3-144*n^2-690*n+78+(1944*n^3+1080*n^2-2862*n+396)*x+(3888*n^2-1890*n+270)*x^2+(1944*n-648)*x^3)*P(n-1,x) - (180*n^2+150*n-5+(972*n^2+1170*n+18)*x+(1296*n^2+3024*n+387)*x^2+(2592*n+1296)*x^3+1296*x^4)*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[((864*n^3 - 144*n^2 - 690*n + 78 + (1944*n^3 + 1080*n^2 - 2862*n + 396)*x + (3888*n^2 - 1890*n + 270)*x^2 + (1944*n - 648)*x^3)*P[n - 1, x] - (180*n^2 + 150*n - 5 + (972*n^2 + 1170*n + 18)*x + (1296*n^2 + 3024*n + 387)*x^2 + (2592*n + 1296)*x^3 + 1296*x^4)*D[P[n - 1, x], x])/(216*n^3 - 144*n^2 - 60*n + (432*n^2 - 378*n)*x + 216*n*x^2)]];
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] = ((864*n^3 - 144*n^2 - 690*n + 78 + (1944*n^3 + 1080*n^2 - 2862*n + 396)*x + (3888*n^2 - 1890*n + 270)*x^2 + (1944*n - 648)*x^3)*v[i - 1] - (180*n^2 + 150*n - 5 + (972*n^2 + 1170*n + 18)*x + (1296*n^2 + 3024*n + 387)*x^2 + (2592*n + 1296)*x^3 + 1296*x^4)*deriv(v[i - 1], x))/(216*n^3 - 144*n^2 - 60*n + (432*n^2 - 378*n)*x + 216*n*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. [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