Real-rooted atlas

A123519 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(33 - 12\,x^{2} + 58\,n - 8\,n x^{2} + 36\,n^{2} + 8\,n^{3}\right)\,P_{n}(x) = \left(165 + 152\,x - 60\,x^{2} - 32\,x^{3} + 236\,n + 236\,n x - 32\,n x^{2} - 16\,n x^{3} + 108\,n^{2} + 112\,n^{2} x + 16\,n^{3} + 16\,n^{3} x\right)\,P_{n-1}(x) + \left(33 + 64\,x - 32\,x^{2} - 64\,x^{3} - 16\,x^{4} + 48\,n + 128\,n x + 64\,n x^{2} + 12\,n^{2} + 32\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 39 + 60\,x + 16\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0396016115
1193504368641129
289733364176198425610649
340071940434864284169984102497699
417415104052244832286976172800481284096878299
574177527600153820823660802049280976896225280163847773905
631110525679528941152170735361941888013309952524697610321926553667967281
71288967121143244907126411208985615839462414257868880769024271319044653056262144588353851

All 15 rows

Technical details
Node id
n-fd3d2e7c65be4d2d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-0b9616e9ad3f4864

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:       39,        60,         16;
  1:      193,       504,        368,         64;
  2:      897,      3336,       4176,       1984,        256;
  3:     4007,     19404,      34864,      28416,       9984,        1024;
  4:    17415,    104052,     244832,     286976,     172800,       48128,       4096;
  5:    74177,    527600,    1538208,    2366080,    2049280,      976896,     225280,      16384;
  6:   311105,   2567952,    8941152,   17073536,   19418880,    13309952,    5246976,    1032192,      65536;
  7:  1288967,  12114324,   49071264,  112089856,  158394624,   142578688,   80769024,   27131904,    4653056,    262144;
  8:  5287847,  55757996,  257673456,  685708544, 1160074496,  1297787904,  965328896,  465698816,  136249344,  20709376,  1048576;
  9: 21516033, 251566824, 1306421648, 3971080384, 7833035520, 10495270912, 9702490112, 6144196608, 2580086784, 668467200, 91226112, 4194304;
%F Formula
The row polynomials satisfy (8*n^3+36*n^2+58*n+33-(8*n+12)*x^2)*P(n,x) = (16*n^3+108*n^2+236*n+165+(16*n^3+112*n^2+236*n+152)*x-(32*n+60)*x^2-(16*n+32)*x^3)*P(n-1,x) + (12*n^2+48*n+33+(32*n^2+128*n+64)*x+(16*n^2+64*n-32)*x^2-64*x^3-16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 39 + 60*x + 16*x^2.
%t Mathematica
P[0, x_] := 39 + 60*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^3 + 108*n^2 + 236*n + 165 + (16*n^3 + 112*n^2 + 236*n + 152)*x - (32*n + 60)*x^2 - (16*n + 32)*x^3)*P[n - 1, x] + (12*n^2 + 48*n + 33 + (32*n^2 + 128*n + 64)*x + (16*n^2 + 64*n - 32)*x^2 - 64*x^3 - 16*x^4)*D[P[n - 1, x], x])/(8*n^3 + 36*n^2 + 58*n + 33 - (8*n + 12)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 39 + 60*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^3 + 108*n^2 + 236*n + 165 + (16*n^3 + 112*n^2 + 236*n + 152)*x - (32*n + 60)*x^2 - (16*n + 32)*x^3)*v[i - 1] + (12*n^2 + 48*n + 33 + (32*n^2 + 128*n + 64)*x + (16*n^2 + 64*n - 32)*x^2 - 64*x^3 - 16*x^4)*deriv(v[i - 1], x))/(8*n^3 + 36*n^2 + 58*n + 33 - (8*n + 12)*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