Real-rooted atlas

A094816 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 + 7\,n - n^{2}\right)\,P_{n-4}(x) + \left(9 - 24\,x - 9\,n + 8\,n x + 2\,n^{2}\right)\,P_{n-3}(x) + \left(4 + 8\,x - 16\,x^{2} - 8\,n x - n^{2}\right)\,P_{n-2}(x) + \left(-1 + 8\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 4\)
\(P_{3}(x) = 20 + 32\,x\)
\(P_{4}(x) = 92 + 288\,x + 192\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
244
3203252
492288192572
54642080268810246256
62676150402784020480512071156
71771611628827168028672013824024576855220
813350098179227041283619840250880086016011468810922908
911322889090112283261444499353639710720196116485046272524288148435008
1010693284921050883161690885720494086020044803798466561410662402831155223592962144605092
111113811721016891232377555520075738368009124899840689142988832789299209515827201533542401048576032888346772
12126909762812172693280482463189121053018419201413479372801226982113287009650278426125271040609878016080740352046137344534210195196
131570709672015721199011265870719833615429254410242263658496000219774111744014477779599366496022691841951609651203748659200041523609602013265929170332813424
142098784279242180552315968958621689043223850439129088377096709017604016155788902429683858620416153993420472325582278361088138250551296022252041011220937965568872415232165990630886804
15301169135144432343556291456148313373673920388825046285312655469698723840754859154137088613857369309184358242940616704150494122082304450603830476809366849716224128245039104010381741260837580963843161234211135188
Technical details
Node id
n-7007ff45078bac1e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6e6ab4cefb774a1e

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        0;
   2:        4;
   3:       20,       32;
   4:       92,      288,       192;
   5:      464,     2080,      2688,      1024;
   6:     2676,    15040,     27840,     20480,      5120;
   7:    17716,   116288,    271680,    286720,    138240,     24576;
   8:   133500,   981792,   2704128,   3619840,   2508800,    860160,    114688;
   9:  1132288,  9090112,  28326144,  44993536,  39710720,  19611648,   5046272,   524288;
  10: 10693284, 92105088, 316169088, 572049408, 602004480, 379846656, 141066240, 28311552, 2359296;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-9*n+9+(8*n-24)*x)*P(n-3,x) - (n^2-4+(8*n-8)*x+16*x^2)*P(n-2,x) + (2*n-1+8*x)*P(n-1,x) for n >= 5, with P(1,x) = 0, P(2,x) = 4, P(3,x) = 20 + 32*x, P(4,x) = 92 + 288*x + 192*x^2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 4;
P[3, x_] := 20 + 32*x;
P[4, x_] := 92 + 288*x + 192*x^2;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 9*n + 9 + (8*n - 24)*x)*P[n - 3, x] - (n^2 - 4 + (8*n - 8)*x + 16*x^2)*P[n - 2, x] + (2*n - 1 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; v[3] = 20 + 32*x; v[4] = 92 + 288*x + 192*x^2; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 9*n + 9 + (8*n - 24)*x)*v[i - 3] - (n^2 - 4 + (8*n - 8)*x + 16*x^2)*v[i - 2] + (2*n - 1 + 8*x)*v[i - 1]); 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