Real-rooted atlas

A123519 → basis A007318 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−4, −2)coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(3 + 2\,n\right)\,P_{n}(x) = \left(11 + 4\,x + 6\,n + 2\,n x\right)\,P_{n-1}(x) + \left(16 + 12\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 29 + 22\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
02922455
1169188688433
29851444780184163409
3574110446750026564643226839
4334617271465112307528080112064211303
51950254927605286003120801095202284826241281663585
61136689327389640920082898928127320035500861376601625613097377
76625109214239943057124025265504133275364653600107564815872013568512103115431
83861396513852028622218342020983657612922302454221664156997123097600398592302081024811826071
92250586818869256521579569852167812079211816837605791224322015659524982630485731849779206656020486391493137
101311738121563315098811029086996130180278801031712112057838804482351940864698992128150693120229836802354176145408409650320119025
117645370045355354565187586125410098488220480867570987205475840249625477840128884800819222935697924383104006002585655787523153928192396169459063
1244560482149222870328210515209350640729633803200707201912160496280731968260031904896103344737280313067420167194018304123442176015337881613045760679936163843119035553479
1325971752284913908074770403461135506160530973330000056157219806404337329813632252727505318411312842252803922363568641053392834562173454131233835376643847168003016294414581763276824556114968769
14151374465494586414095797282303321864040038051037009120436065965716803675914211110423578278041472117389107207684584679849728140931439616033985139302463616929792906233036894973132869058560311296065536193329884196673

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-1\right)\,P_{n-2}(x) + \left(6 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 29 + 22\,x + 4\,x^{2}\)
\(P_{1}(x) = 169 + 188\,x + 68\,x^{2} + 8\,x^{3}\)
Technical details
Node id
n-7967b266138e041c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-8d5faac942c17aea)
Fingerprint
f6-f5bc6b0dc876ed19

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:        29,        22,          4;
  1:       169,       188,         68,          8;
  2:       985,      1444,        780,        184,         16;
  3:      5741,     10446,       7500,       2656,        464,        32;
  4:     33461,     72714,      65112,      30752,       8080,      1120,        64;
  5:    195025,    492760,     528600,     312080,     109520,     22848,      2624,      128;
  6:   1136689,   3273896,    4092008,    2898928,    1273200,    355008,     61376,     6016,     256;
  7:   6625109,  21423994,   30571240,   25265504,   13327536,   4653600,   1075648,   158720,   13568,    512;
  8:  38613965, 138520286,  222183420,  209836576,  129223024,  54221664,  15699712,  3097600,  398592,  30208,  1024;
  9: 225058681, 886925652, 1579569852, 1678120792, 1181683760, 579122432, 201565952, 49826304, 8573184, 977920, 66560, 2048;
%F Formula
The row polynomials satisfy (2*n+3)*P(n,x) = (6*n+11+(2*n+4)*x)*P(n-1,x) + (16+12*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 29 + 22*x + 4*x^2.
%t Mathematica
P[0, x_] := 29 + 22*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 11 + (2*n + 4)*x)*P[n - 1, x] + (16 + 12*x + 2*x^2)*D[P[n - 1, x], x])/(2*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 29 + 22*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 11 + (2*n + 4)*x)*v[i - 1] + (16 + 12*x + 2*x^2)*deriv(v[i - 1], x))/(2*n + 3)); 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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023