Real-rooted atlas

A207824 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-6 + 6\,n\right)\,P_{n}(x) = \left(36 + 18\,x + 18\,n + 9\,n x\right)\,P_{n-1}(x) + \left(32 + 36\,x + 9\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2361854
331832481723
42484383419443248586
51815337584288369720121595508
61272243305883402001733404374043741019466
78663642709288350001023911209112501837081530910577049
8577720821123324328516562817342014390460437837473483252488107481762
937913199158693904288664884298056024191081835778921921971799228343521771471075031529
1024569362811585192942413260216291482226022498106401150869384390195792845581681062882059049010618948692
11157608210682665836281940423323526843918400242383654801492664241663494996041842328800348979590389723401948617103837554216
12100258085885789747511015121176552023576562108024384507624017566710084089950050288327391275608301108420139650885014030042463772921006946320212
136332899336539927311856011484049978801992596870160232295317101019169182370881148185655064502992608976159952226445360104421605448333132497428776207261999696582808815
143976344856802717715196440853594725528016313435591400211661122129201968847373732413508377709904692185305794426490123055447478402322601513841574962080910379617410007166696156692820403008270
Technical details
Node id
n-220b8d2b1d2106bd · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-565a6fbdecfd996d

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:        3;
  2:       36,        18;
  3:      318,       324,        81;
  4:     2484,      3834,      1944,       324;
  5:    18153,     37584,     28836,      9720,      1215;
  6:   127224,    330588,    340200,    173340,     43740,     4374;
  7:   866364,   2709288,   3500010,   2391120,    911250,   183708,    15309;
  8:  5777208,  21123324,  32851656,  28173420,  14390460,  4378374,   734832,   52488;
  9: 37913199, 158693904, 288664884, 298056024, 191081835, 77892192, 19717992, 2834352, 177147;
%F Formula
The row polynomials satisfy (6*n-6)*P(n,x) = (18*n+36+(9*n+18)*x)*P(n-1,x) + (32+36*x+9*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((18*n + 36 + (9*n + 18)*x)*P[n - 1, x] + (32 + 36*x + 9*x^2)*D[P[n - 1, x], x])/(6*n - 6)]];
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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((18*n + 36 + (9*n + 18)*x)*v[i - 1] + (32 + 36*x + 9*x^2)*deriv(v[i - 1], x))/(6*n - 6)); 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