Real-rooted atlas

A090452 → D

nonnegative coefficients

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

Recurrence

\(\left(-8 - 12\,x + 2\,n + 6\,n x + 5\,n^{2} + 6\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(10\,n + 54\,n x + 92\,n x^{2} + 48\,n x^{3} + 7\,n^{2} + 23\,n^{2} x + 16\,n^{2} x^{2} + n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(7\,n x + 26\,n x^{2} + 31\,n x^{3} + 12\,n x^{4} + 2\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
1347
27324520104
312102315456315841284
4182361188308445503864176433615040
5254603330129403041545480436802616089101320172720
6338047800414601374153027844573174770003385801564204247151481967232
7421302161701113004859401444506304145146295205124735409332023012998648641951952002022329664

All 15 rows

Technical details
Node id
n-97c554796bb8ac7a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-60f5152abc48001b

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  0;
  1:  3,    4;
  2:  7,   32,    45,     20;
  3: 12,  102,   315,    456,     315,       84;
  4: 18,  236,  1188,   3084,    4550,     3864,     1764,       336;
  5: 25,  460,  3330,  12940,   30415,    45480,    43680,     26160,      8910,      1320;
  6: 33,  804,  7800,  41460,  137415,   302784,   457317,    477000,    338580,    156420,     42471,      5148;
  7: 42, 1302, 16170, 111300,  485940,  1444506,  3041451,   4629520,   5124735,   4093320,   2301299,    864864,    195195,     20020;
  8: 52, 1992, 30660, 263312, 1449420,  5510904, 15129016,  30797720,  47169540,  54620720,  47611564,  30774744,  14314300,   4532528,   875160,    77792;
  9: 63, 2916, 54288, 566208, 3810240, 17875872, 61299756, 158351256, 314061165, 483449220, 579958236, 540874152, 388381266, 210686112, 83538000, 22849632, 3854682, 302328;
%F Formula
The row polynomials satisfy (n^3+5*n^2+2*n-8+(6*n^2+6*n-12)*x)*P(n,x) = (n^3+7*n^2+10*n+(n^3+23*n^2+54*n)*x+(16*n^2+92*n)*x^2+48*n*x^3)*P(n-1,x) + ((2*n^2+7*n)*x+(4*n^2+26*n)*x^2+(2*n^2+31*n)*x^3+12*n*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + 7*n^2 + 10*n + (n^3 + 23*n^2 + 54*n)*x + (16*n^2 + 92*n)*x^2 + 48*n*x^3)*P[n - 1, x] + ((2*n^2 + 7*n)*x + (4*n^2 + 26*n)*x^2 + (2*n^2 + 31*n)*x^3 + 12*n*x^4)*D[P[n - 1, x], x])/(n^3 + 5*n^2 + 2*n - 8 + (6*n^2 + 6*n - 12)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 + 7*n^2 + 10*n + (n^3 + 23*n^2 + 54*n)*x + (16*n^2 + 92*n)*x^2 + 48*n*x^3)*v[i - 1] + ((2*n^2 + 7*n)*x + (4*n^2 + 26*n)*x^2 + (2*n^2 + 31*n)*x^3 + 12*n*x^4)*deriv(v[i - 1], x))/(n^3 + 5*n^2 + 2*n - 8 + (6*n^2 + 6*n - 12)*x)); 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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.