Real-rooted atlas

A123519 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(6 + 10\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(42 + 96\,x + 38\,n + 80\,n x + 8\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(6 + 32\,x + 32\,x^{2} + 3\,n + 16\,n x + 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 56 + 128\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
056128184
1304140815363248
2148810368230401638451280
3687263872215040311296163840760920
430568355072160358435225603768320157286410852968
513244818455041046016030883840501350404246732814680064150604384
65628489146112623831042315878405056102406495928324550819841342177282048182688
723563924374451234882560015608872964288184320740976230478685143044697620480120795955227427854760

All 15 rows

Technical details
Node id
n-8bfdefa57f60afcf · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-230a3cc4b5b885b4

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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:       56,       128;
  1:      304,      1408,       1536;
  2:     1488,     10368,      23040,       16384;
  3:     6872,     63872,     215040,      311296,       163840;
  4:    30568,    355072,    1603584,     3522560,      3768320,      1572864;
  5:   132448,   1845504,   10460160,    30883840,     50135040,     42467328,      14680064;
  6:   562848,   9146112,   62383104,   231587840,    505610240,    649592832,     455081984,     134217728;
  7:  2356392,  43744512,  348825600,  1560887296,   4288184320,   7409762304,    7868514304,    4697620480,    1207959552;
  8:  9748888, 203534208, 1857853440,  9732767744,  32256000000,  70156025856,  100176756736,   90596966400,   47110422528,  10737418240;
  9: 39946384, 926374528, 9524998656, 57187287040, 222063493120, 582871941120, 1047628087296, 1273592020992, 1002606428160, 461708984320, 94489280512;
%F Formula
The row polynomials satisfy (4*n^2+10*n+6)*P(n,x) = (8*n^2+38*n+42+(16*n^2+80*n+96)*x)*P(n-1,x) + (3*n+6+(16*n+32)*x+(16*n+32)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 56 + 128*x.
%t Mathematica
P[0, x_] := 56 + 128*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 + 38*n + 42 + (16*n^2 + 80*n + 96)*x)*P[n - 1, x] + (3*n + 6 + (16*n + 32)*x + (16*n + 32)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 10*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 56 + 128*x; for(n = 1, N, my(i = n + 1); v[i] = ((8*n^2 + 38*n + 42 + (16*n^2 + 80*n + 96)*x)*v[i - 1] + (3*n + 6 + (16*n + 32)*x + (16*n + 32)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 10*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