Real-rooted atlas

A082985 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(5 + x - 4\,x^{2} - 14\,n - 10\,n x + 8\,n x^{2} + 4\,n^{2} + 16\,n^{2} x + 8\,n^{3}\right)\,P_{n}(x) = \left(23 + 6\,x - 25\,x^{2} - 12\,x^{3} - 74\,n - 64\,n x + 26\,n x^{2} + 24\,n x^{3} + 40\,n^{2} + 92\,n^{2} x + 48\,n^{2} x^{2} + 32\,n^{3} + 24\,n^{3} x\right)\,P_{n-1}(x) + \left(-5 - 44\,x - 87\,x^{2} - 64\,x^{3} - 16\,x^{4} - 50\,n - 130\,n x - 112\,n x^{2} - 32\,n x^{3} - 20\,n^{2} - 36\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17310
22625556
385126567274
426553136910291276
580420351936847165115798
6239373328892538216772471325936
7701925275373502904012630300335015114682
820350842691470671401147879526350499847617502436
958445273714551380623029431433187625503887866627192184526
1016654187055519891832603006214588511541394051259007211844805219437176
1147143227213836957086103538419914426636964327759628103591525591720410122340544930
121326961838500023718600395787004320615032329690168822506140400152405024722524255125025173364556
1337168152552021179151580146406960179558100153509850935942404090419012690405272415038610033345152127738197494
14103663707685664125937800152684706071742897069034210048187734024645186092112555247896354665375584350448631827293131746976
Technical details
Node id
n-c4395efdeea4954c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-346f457e30c539a6

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,      3;
  2:    26,     25,      5;
  3:    85,    126,     56,      7;
  4:   265,    531,    369,    102,      9;
  5:   804,   2035,   1936,    847,    165,     11;
  6:  2393,   7332,   8892,   5382,   1677,    247,    13;
  7:  7019,  25275,  37350,  29040,  12630,   3003,   350,   15;
  8: 20350,  84269, 147067, 140114,  78795,  26350,  4998,  476,  17;
  9: 58445, 273714, 551380, 623029, 431433, 187625, 50388, 7866, 627, 19;
%F Formula
The row polynomials satisfy (8*n^3+4*n^2-14*n+5+(16*n^2-10*n+1)*x+(8*n-4)*x^2)*P(n,x) = (32*n^3+40*n^2-74*n+23+(24*n^3+92*n^2-64*n+6)*x+(48*n^2+26*n-25)*x^2+(24*n-12)*x^3)*P(n-1,x) - (20*n^2+50*n+5+(36*n^2+130*n+44)*x+(16*n^2+112*n+87)*x^2+(32*n+64)*x^3+16*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[((32*n^3 + 40*n^2 - 74*n + 23 + (24*n^3 + 92*n^2 - 64*n + 6)*x + (48*n^2 + 26*n - 25)*x^2 + (24*n - 12)*x^3)*P[n - 1, x] - (20*n^2 + 50*n + 5 + (36*n^2 + 130*n + 44)*x + (16*n^2 + 112*n + 87)*x^2 + (32*n + 64)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 + 4*n^2 - 14*n + 5 + (16*n^2 - 10*n + 1)*x + (8*n - 4)*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] = ((32*n^3 + 40*n^2 - 74*n + 23 + (24*n^3 + 92*n^2 - 64*n + 6)*x + (48*n^2 + 26*n - 25)*x^2 + (24*n - 12)*x^3)*v[i - 1] - (20*n^2 + 50*n + 5 + (36*n^2 + 130*n + 44)*x + (16*n^2 + 112*n + 87)*x^2 + (32*n + 64)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 + 4*n^2 - 14*n + 5 + (16*n^2 - 10*n + 1)*x + (8*n - 4)*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