Real-rooted atlas

A172431 → basis A007318 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(-4 - 2\,x + 14\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-18 - 17\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1527
22116340
38594344217
43414922616051159
51365242616925769566160
6546111528995944881106140732689
7218455339855014308601013019321968173383
887381242724290457195012800352044831502649919480
9349525108741014822401159080571225183202380044872345104875913
101398101481568073659156575280378364014615163819976626472274401125856071
115592405211268623583281035955540236653001069807233769687406081097911036255012137109280
1222369621919639961712932931908628601414359857326505627123292720241613542101744601444367613727060321
13894784853976821548069633649886206248145894754757882222021811126309028814397318235950026769619656819143855438727
14357913941171003327237547823515016441240454969995029584530361419492627509245776136817370272750403947658398736262089801520444528200

Other recurrences

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

\(P_{n}(x) = \left(-4 - 4\,x - x^{2}\right)\,P_{n-2}(x) + \left(5 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 2\,x\)
Technical details
Node id
n-2694e6ea2b3679ca · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-4abba1b6f6f81fd4)
Fingerprint
f6-afc5f7338fb19d0f

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:      5,       2;
  2:     21,      16,       3;
  3:     85,      94,      34,       4;
  4:    341,     492,     261,      60,      5;
  5:   1365,    2426,    1692,     576,     95,      6;
  6:   5461,   11528,    9959,    4488,   1106,    140,     7;
  7:  21845,   53398,   55014,   30860,  10130,   1932,   196,    8;
  8:  87381,  242724,  290457,  195012,  80035,  20448,  3150,  264,   9;
  9: 349525, 1087410, 1482240, 1159080, 571225, 183202, 38004, 4872, 345, 10;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (14*n-4+(6*n-2)*x)*P(n-1,x) - (18+17*x+4*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((14*n - 4 + (6*n - 2)*x)*P[n - 1, x] - (18 + 17*x + 4*x^2)*D[P[n - 1, x], x])/(2*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((14*n - 4 + (6*n - 2)*x)*v[i - 1] - (18 + 17*x + 4*x^2)*deriv(v[i - 1], x))/(2*n)); 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