Real-rooted atlas

A085478 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-516 - 96\,x + 192\,x^{2} + 920\,n + 64\,n x - 128\,n x^{2} - 576\,n^{2} + 128\,n^{3}\right)\,P_{n}(x) = \left(-96 + 80\,x + 384\,x^{2} + 256\,x^{3} + 308\,n + 144\,n x - 320\,n x^{2} - 256\,n x^{3} - 448\,n^{2} - 512\,n^{2} x + 192\,n^{3} + 256\,n^{3} x\right)\,P_{n-1}(x) + \left(15 - 48\,x - 608\,x^{2} - 768\,x^{3} - 256\,x^{4} - 160\,n - 768\,n x - 512\,n x^{2} + 80\,n^{2} + 384\,n^{2} x + 256\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
111
26410
325521693
48933232064805
52941604278417282566666
6929669617136190728704102453561
728492556087120142976116480419844096421065
88550918443923528592001030400660480196608163843255818
925241315932162512044712327153920673689635553289011206553624850325
10735611051492633131221026944422945285309337641086976184156164063232262144187699181
1121216634103082353900891721856222917632351740928362471424237830144926023681808793610485761405582346
1260672110832976843451203775406081077756160205525196826608762882322432000132179558445481984079691776419430410450143345
131722625338277602933380801483904000487164800010917017600170793369601871659008014155776000711065600021915238403481272321677721677220245393
1448612541041312049953404805617187712208642918405380141363298920046592130950758400124269428736828742041603724437094410393485312150994944067108864567616578570
Technical details
Node id
n-c60b7a28f15389b9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2a10791ffa3613cc

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:     1;
  2:     6,      4;
  3:    25,     52,      16;
  4:    89,    332,     320,      64;
  5:   294,   1604,    2784,    1728,     256;
  6:   929,   6696,   17136,   19072,    8704,    1024;
  7:  2849,  25560,   87120,  142976,  116480,   41984,    4096;
  8:  8550,  91844,  392352,  859200, 1030400,  660480,  196608,  16384;
  9: 25241, 315932, 1625120, 4471232, 7153920, 6736896, 3555328, 901120, 65536;
%F Formula
The row polynomials satisfy (128*n^3-576*n^2+920*n-516+(64*n-96)*x-(128*n-192)*x^2)*P(n,x) = (192*n^3-448*n^2+308*n-96+(256*n^3-512*n^2+144*n+80)*x-(320*n-384)*x^2-(256*n-256)*x^3)*P(n-1,x) + (80*n^2-160*n+15+(384*n^2-768*n-48)*x+(256*n^2-512*n-608)*x^2-768*x^3-256*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[((192*n^3 - 448*n^2 + 308*n - 96 + (256*n^3 - 512*n^2 + 144*n + 80)*x - (320*n - 384)*x^2 - (256*n - 256)*x^3)*P[n - 1, x] + (80*n^2 - 160*n + 15 + (384*n^2 - 768*n - 48)*x + (256*n^2 - 512*n - 608)*x^2 - 768*x^3 - 256*x^4)*D[P[n - 1, x], x])/(128*n^3 - 576*n^2 + 920*n - 516 + (64*n - 96)*x - (128*n - 192)*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] = ((192*n^3 - 448*n^2 + 308*n - 96 + (256*n^3 - 512*n^2 + 144*n + 80)*x - (320*n - 384)*x^2 - (256*n - 256)*x^3)*v[i - 1] + (80*n^2 - 160*n + 15 + (384*n^2 - 768*n - 48)*x + (256*n^2 - 512*n - 608)*x^2 - 768*x^3 - 256*x^4)*deriv(v[i - 1], x))/(128*n^3 - 576*n^2 + 920*n - 516 + (64*n - 96)*x - (128*n - 192)*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