Real-rooted atlas

A232535 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(6 - 19\,n + 3\,n x + 19\,n^{2} - 9\,n^{2} x - 6\,n^{3} + 6\,n^{3} x\right)\,P_{n}(x) = \left(-12 - 36\,x - 16\,n - 51\,n x - 9\,n x^{2} + 52\,n^{2} + 117\,n^{2} x - 63\,n^{2} x^{2} - 24\,n^{3} - 30\,n^{3} x + 54\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 12\,x^{2} - 8\,n x - 24\,n x^{2} + 12\,n^{2} x + 24\,n^{2} x^{2} - 36\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
166
2425496
32406663241230
4124853286156162014352
561443542466096437407290158694
629184211968549504607824264870306181693968
71351681184256393984061896964626720143904612247217637198
86144006303744256296965246208056687040308629447225848472392180258144
92752512323665921556029443935692805611161604480375681866473283416968817714701816033542
101218969616161177689712230427056332804811788800512394854431685955841046190816154117890649539018087694080
115347737678918451249686773761741691289637205360640498274882564147195852820535825024551848334466902517023383404178479776526
12232783872378431078426648248320106508058624265991454720430662500352454051713024305035061760124025574816276963429602813802948829047961747532756976
13100663296017874026496139207901184625012899840178824609792034026601512964362056810496373570784870420751991603207065945057601333477585441152376664429016678616998727726950
1443285217288334920908871141267865635455972147201144085741568025048153718784379153559715843968612809113628262797655040132293946470403832633449216619796254896461365189741004423490164426945770032
Technical details
Node id
n-d9f1d0e229a4b65a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-286b38aa8cd390f7

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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:       0;
  1:       6;
  2:      42,       54;
  3:     240,      666,       324;
  4:    1248,     5328,      6156,      1620;
  5:    6144,    35424,     66096,     43740,      7290;
  6:   29184,   211968,    549504,    607824,    264870,     30618;
  7:  135168,  1184256,   3939840,   6189696,   4626720,   1439046,    122472;
  8:  614400,  6303744,  25629696,  52462080,  56687040,  30862944,   7225848,   472392;
  9: 2752512, 32366592, 155602944, 393569280, 561116160, 448037568, 186647328, 34169688, 1771470;
%F Formula
The row polynomials satisfy -(6*n^3-19*n^2+19*n-6-(6*n^3-9*n^2+3*n)*x)*P(n,x) = -(24*n^3-52*n^2+16*n+12+(30*n^3-117*n^2+51*n+36)*x-(54*n^3-63*n^2-9*n)*x^2)*P(n-1,x) + ((12*n^2-8*n-4)*x+(24*n^2-24*n-12)*x^2-36*n^2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[(-(24*n^3 - 52*n^2 + 16*n + 12 + (30*n^3 - 117*n^2 + 51*n + 36)*x - (54*n^3 - 63*n^2 - 9*n)*x^2)*P[n - 1, x] + ((12*n^2 - 8*n - 4)*x + (24*n^2 - 24*n - 12)*x^2 - 36*n^2*x^3)*D[P[n - 1, x], x])/(-6*n^3 + 19*n^2 - 19*n + 6 + (6*n^3 - 9*n^2 + 3*n)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = (-(24*n^3 - 52*n^2 + 16*n + 12 + (30*n^3 - 117*n^2 + 51*n + 36)*x - (54*n^3 - 63*n^2 - 9*n)*x^2)*v[i - 1] + ((12*n^2 - 8*n - 4)*x + (24*n^2 - 24*n - 12)*x^2 - 36*n^2*x^3)*deriv(v[i - 1], x))/(-6*n^3 + 19*n^2 - 19*n + 6 + (6*n^3 - 9*n^2 + 3*n)*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