Real-rooted atlas

A086645 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-30 + 42\,x - 8\,x^{2} + 47\,n - 37\,n x + 4\,n x^{2} - 24\,n^{2} + 8\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(-40 - 8\,x + 376\,x^{2} - 96\,x^{3} + 116\,n + 260\,n x - 380\,n x^{2} + 48\,n x^{3} - 80\,n^{2} - 224\,n^{2} x + 96\,n^{2} x^{2} + 16\,n^{3} + 48\,n^{3} x\right)\,P_{n-1}(x) + \left(-8\,x - 6\,x^{2} + 96\,x^{3} - 32\,x^{4} + 20\,n x + 64\,n x^{2} - 64\,n x^{3} - 8\,n^{2} x - 32\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
26410
3406416120
4224768480641536
5115261448704307225619328
65632409609472076800179201024237056
72662424576079872010895365836809830440962846720
812288013762565791744113541121043660840140805160961638433628160

All 15 rows

Technical details
Node id
n-c0564d8d108d2c29 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-5ebdabbafcbfe122

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       6,        4;
   3:      40,       64,        16;
   4:     224,      768,       480,        64;
   5:    1152,     6144,      8704,      3072,        256;
   6:    5632,    40960,     94720,     76800,      17920,       1024;
   7:   26624,   245760,    798720,   1089536,     583680,      98304,      4096;
   8:  122880,  1376256,   5791744,  11354112,   10436608,    4014080,    516096,     16384;
   9:  557056,  7340032,  38010880,  98041856,  131727360,   88080384,  25690112,   2621440,    65536;
  10: 2490368, 37748736, 232390656, 746323968, 1336541184, 1323565056, 677117952, 155713536, 12976128, 262144;
%F Formula
The row polynomials satisfy (4*n^3-24*n^2+47*n-30+(8*n^2-37*n+42)*x+(4*n-8)*x^2)*P(n,x) = (16*n^3-80*n^2+116*n-40+(48*n^3-224*n^2+260*n-8)*x+(96*n^2-380*n+376)*x^2+(48*n-96)*x^3)*P(n-1,x) - ((8*n^2-20*n+8)*x+(32*n^2-64*n+6)*x^2+(64*n-96)*x^3+32*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^3 - 80*n^2 + 116*n - 40 + (48*n^3 - 224*n^2 + 260*n - 8)*x + (96*n^2 - 380*n + 376)*x^2 + (48*n - 96)*x^3)*P[n - 1, x] - ((8*n^2 - 20*n + 8)*x + (32*n^2 - 64*n + 6)*x^2 + (64*n - 96)*x^3 + 32*x^4)*D[P[n - 1, x], x])/(4*n^3 - 24*n^2 + 47*n - 30 + (8*n^2 - 37*n + 42)*x + (4*n - 8)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 6 + 4*x; for(n = 3, N, my(i = n); v[i] = ((16*n^3 - 80*n^2 + 116*n - 40 + (48*n^3 - 224*n^2 + 260*n - 8)*x + (96*n^2 - 380*n + 376)*x^2 + (48*n - 96)*x^3)*v[i - 1] - ((8*n^2 - 20*n + 8)*x + (32*n^2 - 64*n + 6)*x^2 + (64*n - 96)*x^3 + 32*x^4)*deriv(v[i - 1], x))/(4*n^3 - 24*n^2 + 47*n - 30 + (8*n^2 - 37*n + 42)*x + (4*n - 8)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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