Real-rooted atlas

A125662 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(10\,n + 8\,n x - 8\,n x^{2} + 8\,n^{3}\right)\,P_{n}(x) = \left(24 + 20\,x - 16\,x^{2} - 16\,x^{3} + 56\,n + 52\,n x - 16\,n x^{2} - 16\,n x^{3} + 48\,n^{2} + 48\,n^{2} x + 16\,n^{3} + 16\,n^{3} x\right)\,P_{n-1}(x) + \left(3 - 4\,x - 40\,x^{2} - 48\,x^{3} - 16\,x^{4} + 12\,n + 32\,n x + 16\,n x^{2} + 12\,n^{2} + 32\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18412
2476416127
3240568384641256
4113739044560204825611905
55144233083942430464102401024109604
622559127360284256317440183040491524096987903
7967366549601822464268352022323201026048229376163848761808
8407841322291210763616197570562165120014336000547635210485766553676729089
91696808153372365983488013184691218004582415691468886245376281804804718592262144665082940
106984655710952323175083368175554561340120320145470259210515742724938792961409679362097152010485765716408191
112850068832266205616239299844788635840917315584011949973504107540643846638075904272105472068943872092274688419430448785960632
121154483371439128256806139768026800435200588037504008951699865696651362304742360350723996942336014533263360331035443240265318416777216413857027457
1346475474463259755803904126464014451952896035757881344062350897766478656805273672461962444848525528268823162388480075665244160156531425281744830464671088643492636485716
14186098873527465393920185188713920755570708480208207715584040943997091845916068487168636678045696051123067944963033899466752130040410931238560753254473064775680751619276826843545629342478921215
Technical details
Node id
n-711cbd693420f053 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-afdf9d915441c1e3

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:       8,        4;
  2:      47,       64,       16;
  3:     240,      568,      384,        64;
  4:    1137,     3904,     4560,      2048,       256;
  5:    5144,    23308,    39424,     30464,     10240,      1024;
  6:   22559,   127360,   284256,    317440,    183040,     49152,     4096;
  7:   96736,   654960,  1822464,   2683520,   2232320,   1026048,   229376,    16384;
  8:  407841,  3222912, 10763616,  19757056,  21651200,  14336000,  5476352,  1048576,   65536;
  9: 1696808, 15337236, 59834880, 131846912, 180045824, 156914688, 86245376, 28180480, 4718592, 262144;
%F Formula
The row polynomials satisfy (8*n^3+10*n+8*n*x-8*n*x^2)*P(n,x) = (16*n^3+48*n^2+56*n+24+(16*n^3+48*n^2+52*n+20)*x-(16*n+16)*x^2-(16*n+16)*x^3)*P(n-1,x) + (12*n^2+12*n+3+(32*n^2+32*n-4)*x+(16*n^2+16*n-40)*x^2-48*x^3-16*x^4)*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[((16*n^3 + 48*n^2 + 56*n + 24 + (16*n^3 + 48*n^2 + 52*n + 20)*x - (16*n + 16)*x^2 - (16*n + 16)*x^3)*P[n - 1, x] + (12*n^2 + 12*n + 3 + (32*n^2 + 32*n - 4)*x + (16*n^2 + 16*n - 40)*x^2 - 48*x^3 - 16*x^4)*D[P[n - 1, x], x])/(8*n^3 + 10*n + 8*n*x - 8*n*x^2)]];
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] = ((16*n^3 + 48*n^2 + 56*n + 24 + (16*n^3 + 48*n^2 + 52*n + 20)*x - (16*n + 16)*x^2 - (16*n + 16)*x^3)*v[i - 1] + (12*n^2 + 12*n + 3 + (32*n^2 + 32*n - 4)*x + (16*n^2 + 16*n - 40)*x^2 - 48*x^3 - 16*x^4)*deriv(v[i - 1], x))/(8*n^3 + 10*n + 8*n*x - 8*n*x^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