Real-rooted atlas

A172431 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-36\,n - 88\,n x + 64\,n x^{2} - 16\,n^{2} + 128\,n^{2} x + 64\,n^{3}\right)\,P_{n}(x) = \left(28 + 80\,x - 16\,x^{2} - 128\,x^{3} - 264\,n - 680\,n x - 144\,n x^{2} + 384\,n x^{3} + 64\,n^{2} + 544\,n^{2} x + 768\,n^{2} x^{2} + 256\,n^{3} + 384\,n^{3} x\right)\,P_{n-1}(x) + \left(-5 - 68\,x - 276\,x^{2} - 448\,x^{3} - 256\,x^{4} - 100\,n - 520\,n x - 896\,n x^{2} - 512\,n x^{3} - 80\,n^{2} - 288\,n^{2} x - 256\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17411
228441284
39725218432565
431511481404640803587
5984463280406208200019222056
62993173043912043296238565824448132841
789276126817114424848019792083328161281024788219
826228208516694548125254413170408075522714884300823044623228
97614568840426653525754560755112061462403028992838656111360512026865949
10218931221863697953002465910438878000396574082607404810644480248371228160011264154922483
1162443270118883477864010008716818451836822688601618766182410267136035506944710758469836824576887577168
1217689932180472012006096038895840082161360011844102401184462720818088960380723200113484800197683201703936532485056902097
13498264766887716404899920145869536034752592005752373760676050816056612659203336802560134351360034997248053673984410009611468828673050091
141396462020281084413388455325310377600140913832002635342080035633015040351457843202518047360012879027200454744576010469621761427578889748480245760161896262820
Technical details
Node id
n-c9c78e8491065d8d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7bf9c4f1fc46f175

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:     7,      4;
  2:    28,     44,      12;
  3:    97,    252,     184,      32;
  4:   315,   1148,    1404,     640,      80;
  5:   984,   4632,    8040,    6208,    2000,     192;
  6:  2993,  17304,   39120,   43296,   23856,    5824,     448;
  7:  8927,  61268,  171144,  248480,  197920,   83328,   16128,   1024;
  8: 26228, 208516,  694548, 1252544, 1317040,  807552,  271488,  43008,   2304;
  9: 76145, 688404, 2665352, 5754560, 7551120, 6146240, 3028992, 838656, 111360, 5120;
%F Formula
The row polynomials satisfy (64*n^3-16*n^2-36*n+(128*n^2-88*n)*x+64*n*x^2)*P(n,x) = (256*n^3+64*n^2-264*n+28+(384*n^3+544*n^2-680*n+80)*x+(768*n^2-144*n-16)*x^2+(384*n-128)*x^3)*P(n-1,x) - (80*n^2+100*n+5+(288*n^2+520*n+68)*x+(256*n^2+896*n+276)*x^2+(512*n+448)*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[((256*n^3 + 64*n^2 - 264*n + 28 + (384*n^3 + 544*n^2 - 680*n + 80)*x + (768*n^2 - 144*n - 16)*x^2 + (384*n - 128)*x^3)*P[n - 1, x] - (80*n^2 + 100*n + 5 + (288*n^2 + 520*n + 68)*x + (256*n^2 + 896*n + 276)*x^2 + (512*n + 448)*x^3 + 256*x^4)*D[P[n - 1, x], x])/(64*n^3 - 16*n^2 - 36*n + (128*n^2 - 88*n)*x + 64*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] = ((256*n^3 + 64*n^2 - 264*n + 28 + (384*n^3 + 544*n^2 - 680*n + 80)*x + (768*n^2 - 144*n - 16)*x^2 + (384*n - 128)*x^3)*v[i - 1] - (80*n^2 + 100*n + 5 + (288*n^2 + 520*n + 68)*x + (256*n^2 + 896*n + 276)*x^2 + (512*n + 448)*x^3 + 256*x^4)*deriv(v[i - 1], x))/(64*n^3 - 16*n^2 - 36*n + (128*n^2 - 88*n)*x + 64*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