Real-rooted atlas

A111125 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-24 + 51\,x - 12\,x^{2} + 52\,n - 58\,n x + 8\,n x^{2} - 36\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(36\,x + 105\,x^{2} - 36\,x^{3} + 24\,n + 36\,n x - 142\,n x^{2} + 24\,n x^{3} - 28\,n^{2} - 76\,n^{2} x + 48\,n^{2} x^{2} + 8\,n^{3} + 24\,n^{3} x\right)\,P_{n-1}(x) + \left(9\,x^{2} + 32\,x^{3} - 16\,x^{4} + 6\,n x + 16\,n x^{2} - 32\,n x^{3} - 4\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2437
3615526
483535785
51063117669265
6129927529711011804
714143533884637169132393
816195915207523401218245157019
918255144541826545540621423401720350
10203232147758115314177651128635344561958445
112239930451271231605477754304321780554459521166541
1224483416320079594321114121318599538139468834975923471432
13265755525302501040252335803451003302001966256792512155950251326961
1428675715543857171990450927802332963900763776381810111969171991170273716815
153078390776159627146981504517009372472888247288816535227049901779442375114212910366370
Technical details
Node id
n-a33841ee0ddb033c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d672f84bcb723328

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 -1)
  • 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:  1;
   2:  4,   3;
   3:  6,  15,    5;
   4:  8,  35,   35,    7;
   5: 10,  63,  117,   66,     9;
   6: 12,  99,  275,  297,   110,    11;
   7: 14, 143,  533,  884,   637,   169,    13;
   8: 16, 195,  915, 2075,  2340,  1218,   245,   15;
   9: 18, 255, 1445, 4182,  6545,  5406,  2142,  340,  17;
  10: 20, 323, 2147, 7581, 15314, 17765, 11286, 3534, 456, 19;
%F Formula
The row polynomials satisfy (8*n^3-36*n^2+52*n-24+(16*n^2-58*n+51)*x+(8*n-12)*x^2)*P(n,x) = (8*n^3-28*n^2+24*n+(24*n^3-76*n^2+36*n+36)*x+(48*n^2-142*n+105)*x^2+(24*n-36)*x^3)*P(n-1,x) - ((4*n^2-6*n)*x+(16*n^2-16*n-9)*x^2+(32*n-32)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 28*n^2 + 24*n + (24*n^3 - 76*n^2 + 36*n + 36)*x + (48*n^2 - 142*n + 105)*x^2 + (24*n - 36)*x^3)*P[n - 1, x] - ((4*n^2 - 6*n)*x + (16*n^2 - 16*n - 9)*x^2 + (32*n - 32)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 36*n^2 + 52*n - 24 + (16*n^2 - 58*n + 51)*x + (8*n - 12)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^3 - 28*n^2 + 24*n + (24*n^3 - 76*n^2 + 36*n + 36)*x + (48*n^2 - 142*n + 105)*x^2 + (24*n - 36)*x^3)*v[i - 1] - ((4*n^2 - 6*n)*x + (16*n^2 - 16*n - 9)*x^2 + (32*n - 32)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 36*n^2 + 52*n - 24 + (16*n^2 - 58*n + 51)*x + (8*n - 12)*x^2)); 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

Same fingerprint ?

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