Real-rooted atlas

A091042 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-30 + 75\,x - 18\,x^{2} + 71\,n - 86\,n x + 12\,n x^{2} - 52\,n^{2} + 24\,n^{2} x + 12\,n^{3}\right)\,P_{n}(x) = \left(24 + 273\,x + 441\,x^{2} - 162\,x^{3} + 116\,n + 25\,n x - 618\,n x^{2} + 108\,n x^{3} - 160\,n^{2} - 300\,n^{2} x + 216\,n^{2} x^{2} + 48\,n^{3} + 108\,n^{3} x\right)\,P_{n-1}(x) + \left(6\,x + 70\,x^{2} + 132\,x^{3} - 72\,x^{4} + 32\,n x + 48\,n x^{2} - 144\,n x^{3} - 24\,n^{2} x - 72\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
213922
37615045271
4400134410711893004
51984950413608615672931981
6947259136125136104544311852673332146
744032339456958464121305667813214531494773387931
82007041843200653184011232000968112039191046378753280534078648
99011209609216410480648977305610663488067417920208202402676888111537338992921
10399769648562176243044352647184384988505856871098624424173024103716288108453333739773341501710
111756364823946854413748797444322820096810819072092726000646351561216246651609649105148442712110124002932688603751
12765460481157627904750197145627226865664607110266888589282508876948994304423526371841346537649622309499521643333674074381317733228532
13331350016550502400039754137600163676160000423611596800716755599360799497216000579355545600262702440000698037912009795048120620014500132860253071421209221
141426063360258201354242056268021769473611530242794164387840551451525120073955776757766710294347776402895200614415347827123203465420472804178401718423006080894304672129549190254314
15610690662411968865894410421671034885313738571776176044273827843975728608051262467884122112684698552893445172998258073626222709187584852560988364816583906926081739066270768414836794138706101283100306630131
Technical details
Node id
n-31de51a916638825 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3bfd402c4e837e65

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:      13,        9;
   3:      76,      150,        45;
   4:     400,     1344,      1071,       189;
   5:    1984,     9504,     13608,      6156,       729;
   6:    9472,    59136,    125136,    104544,     31185,      2673;
   7:   44032,   339456,    958464,   1213056,    678132,    145314,      9477;
   8:  200704,  1843200,   6531840,  11232000,   9681120,   3919104,    637875,     32805;
   9:  901120,  9609216,  41048064,  89773056, 106634880,  67417920,  20820240,   2676888,   111537;
  10: 3997696, 48562176, 243044352, 647184384, 988505856, 871098624, 424173024, 103716288, 10845333, 373977;
%F Formula
The row polynomials satisfy (12*n^3-52*n^2+71*n-30+(24*n^2-86*n+75)*x+(12*n-18)*x^2)*P(n,x) = (48*n^3-160*n^2+116*n+24+(108*n^3-300*n^2+25*n+273)*x+(216*n^2-618*n+441)*x^2+(108*n-162)*x^3)*P(n-1,x) - ((24*n^2-32*n-6)*x+(72*n^2-48*n-70)*x^2+(144*n-132)*x^3+72*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((48*n^3 - 160*n^2 + 116*n + 24 + (108*n^3 - 300*n^2 + 25*n + 273)*x + (216*n^2 - 618*n + 441)*x^2 + (108*n - 162)*x^3)*P[n - 1, x] - ((24*n^2 - 32*n - 6)*x + (72*n^2 - 48*n - 70)*x^2 + (144*n - 132)*x^3 + 72*x^4)*D[P[n - 1, x], x])/(12*n^3 - 52*n^2 + 71*n - 30 + (24*n^2 - 86*n + 75)*x + (12*n - 18)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((48*n^3 - 160*n^2 + 116*n + 24 + (108*n^3 - 300*n^2 + 25*n + 273)*x + (216*n^2 - 618*n + 441)*x^2 + (108*n - 162)*x^3)*v[i - 1] - ((24*n^2 - 32*n - 6)*x + (72*n^2 - 48*n - 70)*x^2 + (144*n - 132)*x^3 + 72*x^4)*deriv(v[i - 1], x))/(12*n^3 - 52*n^2 + 71*n - 30 + (24*n^2 - 86*n + 75)*x + (12*n - 18)*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

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