Real-rooted atlas

A111125 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-114 + 24\,x + 24\,x^{2} + 148\,n - 16\,n x - 16\,n x^{2} - 72\,n^{2} + 16\,n^{3}\right)\,P_{n}(x) = \left(-24 + 20\,x + 48\,x^{2} + 16\,x^{3} + 62\,n - 12\,n x - 56\,n x^{2} - 16\,n x^{3} - 40\,n^{2} - 32\,n^{2} x + 24\,n^{3} + 16\,n^{3} x\right)\,P_{n-1}(x) + \left(15 - 24\,x - 152\,x^{2} - 96\,x^{3} - 16\,x^{4} - 40\,n - 96\,n x - 32\,n x^{2} + 20\,n^{2} + 48\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2628
32522451
491142648305
5306710552168161752
6979306034921808416329787
7302911988183241380853609926453565
8915043978849368532047920148802304128288616
9271491538183606324565843405601523523942452482561536023
10794395189301434836220270420659681223712454720100864117765128093461
112298661702030543057698281441116236882971844072320129382425113626112102442294584
1265911154576721975759241264384552111204963078430515968127802883545856611840573442048219494007
1318756251717927269620184165031040254744240269552960200191488104875776382924809428480146432012492840961132384889
1453032865324712223893291263432542411112851201355973984118443916874867404834149657611049984024454144345292827033681925812363080
151491264116289683080205583623589719844629134448640917878464523587204793109504263532825610640875523090954246211584080404485816321638429701884283
Technical details
Node id
n-c0d8eaa3461450eb · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-aaa700e4f33e56f7

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:     6,      2;
   3:    25,     22,       4;
   4:    91,    142,      64,       8;
   5:   306,    710,     552,     168,      16;
   6:   979,   3060,    3492,    1808,     416,      32;
   7:  3029,  11988,   18324,   13808,    5360,     992,     64;
   8:  9150,  43978,   84936,   85320,   47920,   14880,   2304,    128;
   9: 27149, 153818,  360632,  456584,  340560,  152352,  39424,   5248,   256;
  10: 79439, 518930, 1434836, 2202704, 2065968, 1223712, 454720, 100864, 11776, 512;
%F Formula
The row polynomials satisfy (16*n^3-72*n^2+148*n-114-(16*n-24)*x-(16*n-24)*x^2)*P(n,x) = (24*n^3-40*n^2+62*n-24+(16*n^3-32*n^2-12*n+20)*x-(56*n-48)*x^2-(16*n-16)*x^3)*P(n-1,x) + (20*n^2-40*n+15+(48*n^2-96*n-24)*x+(16*n^2-32*n-152)*x^2-96*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[((24*n^3 - 40*n^2 + 62*n - 24 + (16*n^3 - 32*n^2 - 12*n + 20)*x - (56*n - 48)*x^2 - (16*n - 16)*x^3)*P[n - 1, x] + (20*n^2 - 40*n + 15 + (48*n^2 - 96*n - 24)*x + (16*n^2 - 32*n - 152)*x^2 - 96*x^3 - 16*x^4)*D[P[n - 1, x], x])/(16*n^3 - 72*n^2 + 148*n - 114 - (16*n - 24)*x - (16*n - 24)*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] = ((24*n^3 - 40*n^2 + 62*n - 24 + (16*n^3 - 32*n^2 - 12*n + 20)*x - (56*n - 48)*x^2 - (16*n - 16)*x^3)*v[i - 1] + (20*n^2 - 40*n + 15 + (48*n^2 - 96*n - 24)*x + (16*n^2 - 32*n - 152)*x^2 - 96*x^3 - 16*x^4)*deriv(v[i - 1], x))/(16*n^3 - 72*n^2 + 148*n - 114 - (16*n - 24)*x - (16*n - 24)*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

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