Real-rooted atlas

A358735 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(n - n^{2}\right)\,P_{n-4}(x) + \left(-2\,n + 4\,n x - 2\,n^{2} - 6\,n^{2} x + 2\,n^{3} + 2\,n^{3} x\right)\,P_{n-3}(x) + \left(-2 + 6\,n - 4\,n x + 2\,n x^{2} + n^{2} + 2\,n^{2} x - 5\,n^{2} x^{2} + 4\,n^{3} x + 4\,n^{3} x^{2} - n^{4} - 2\,n^{4} x - n^{4} x^{2}\right)\,P_{n-2}(x) + \left(4 - 6\,n + 4\,n x - 2\,n^{2} - 6\,n^{2} x + 2\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)
\(P_{2}(x) = 15 + 12\,x + 2\,x^{2}\)
\(P_{3}(x) = 93 + 128\,x + 52\,x^{2} + 6\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
21512229
393128526279
46751368958268243293
5558715243157047520162812045802
6518451797962490121745646428811448720731673
7532705225706439676483726444199944860282491656504013182829
860021253018720464744876769911565515460824128920618944082396840320264262617
9735763514297547491092786520158248955214316993968336695123092687206927624082221123628805831106032
10974830427650048046419176091014328734203483614818962426520686784130605482404219445152840949344902044803628800140408474677
111388183238910424014748835073406356469667718257890551224388880600939084849908599774421356174793661250727296110158194241079340480399168003663088410435
1221144311045517679532742686693168603050151486234746642278637906944023939941166336179555776496169643207060832365618657219294449914649615498003916813989922560479001600102916428090677
133430345368295316399992945031332837938843843391297286113445806588398024247051933094206406234774742003044049997463457761924206145282246560390243392015438258480192233169417446419528103808062270208003097809214603350
145905622261695359613956591535227684661710709367832338304348408150636657695528162080460261599969621239803755225392162446475482791369323118378949952397769991198752012336479162466562668650530638083736835279769629207451571208717829120099450427489502641
Technical details
Node id
n-33c9b9d3f3d7386f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a52fd0bb59d6642d

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 0)
  • 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:        1;
  1:        3,         1;
  2:       15,        12,          2;
  3:       93,       128,         52,          6;
  4:      675,      1368,        958,        268,         24;
  5:     5587,     15243,      15704,       7520,       1628,       120;
  6:    51845,    179796,     249012,     174564,      64288,     11448,       720;
  7:   532705,   2257064,    3967648,    3726444,    1999448,    602824,     91656,     5040;
  8:  6002125,  30187204,   64744876,   76991156,   55154608,  24128920,   6189440,   823968,   40320;
  9: 73576351, 429754749, 1092786520, 1582489552, 1431699396, 833669512, 309268720, 69276240, 8222112, 362880;
%F Formula
The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n)*P(n-4,x) + (2*n^3-2*n^2-2*n+(2*n^3-6*n^2+4*n)*x)*P(n-3,x) - (n^4-n^2-6*n+2+(2*n^4-4*n^3-2*n^2+4*n)*x+(n^4-4*n^3+5*n^2-2*n)*x^2)*P(n-2,x) + (2*n^3-2*n^2-6*n+4+(2*n^3-6*n^2+4*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + x, P(2,x) = 15 + 12*x + 2*x^2, P(3,x) = 93 + 128*x + 52*x^2 + 6*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[2, x_] := 15 + 12*x + 2*x^2;
P[3, x_] := 93 + 128*x + 52*x^2 + 6*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*P[n - 4, x] + (2*n^3 - 2*n^2 - 2*n + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 3, x] - (n^4 - n^2 - 6*n + 2 + (2*n^4 - 4*n^3 - 2*n^2 + 4*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*P[n - 2, x] + (2*n^3 - 2*n^2 - 6*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 1, x])/(n^2 - 3*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; v[3] = 15 + 12*x + 2*x^2; v[4] = 93 + 128*x + 52*x^2 + 6*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*v[i - 4] + (2*n^3 - 2*n^2 - 2*n + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 3] - (n^4 - n^2 - 6*n + 2 + (2*n^4 - 4*n^3 - 2*n^2 + 4*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*v[i - 2] + (2*n^3 - 2*n^2 - 6*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 1])/(n^2 - 3*n + 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