Real-rooted atlas

A172431 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(20\,n + 14\,n x + 20\,n^{2} x + 4\,n^{2} x^{2} + 8\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-20 - 44\,x - 28\,x^{2} - 4\,x^{3} + 80\,n + 126\,n x + 40\,n x^{2} - 8\,n x^{3} + 100\,n^{2} x + 128\,n^{2} x^{2} + 36\,n^{2} x^{3} + 32\,n^{3} x^{2} + 24\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 80\,x - 93\,x^{2} - 46\,x^{3} - 8\,x^{4} - 50\,n x - 130\,n x^{2} - 112\,n x^{3} - 32\,n x^{4} - 20\,n^{2} x^{2} - 36\,n^{2} x^{3} - 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
2820937
321766616179
45526033316025833
514484014221024325363791
637726165508529625555884916989
7987794019998240801565055449806475243

All 15 rows

Technical details
Node id
n-d722a3f2e1a8583f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b9b1d376d6be2c64

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:    3,     4;
  2:    8,    20,      9;
  3:   21,    76,     66,     16;
  4:   55,   260,    333,    160,     25;
  5:  144,   840,   1422,   1024,    325,     36;
  6:  377,  2616,   5508,   5296,   2555,    588,    49;
  7:  987,  7940,  19998,  24080,  15650,   5544,   980,    64;
  8: 2584, 23644,  69327, 100352,  82175,  39528, 10878,  1536,   81;
  9: 6765, 69380, 232134, 392640, 388225, 236580, 89040, 19776, 2295, 100;
%F Formula
The row polynomials satisfy (20*n+(20*n^2+14*n)*x+(8*n^3+4*n^2)*x^2)*P(n,x) = (80*n-20+(100*n^2+126*n-44)*x+(32*n^3+128*n^2+40*n-28)*x^2+(24*n^3+36*n^2-8*n-4)*x^3)*P(n-1,x) - (25+(50*n+80)*x+(20*n^2+130*n+93)*x^2+(36*n^2+112*n+46)*x^3+(16*n^2+32*n+8)*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[((80*n - 20 + (100*n^2 + 126*n - 44)*x + (32*n^3 + 128*n^2 + 40*n - 28)*x^2 + (24*n^3 + 36*n^2 - 8*n - 4)*x^3)*P[n - 1, x] - (25 + (50*n + 80)*x + (20*n^2 + 130*n + 93)*x^2 + (36*n^2 + 112*n + 46)*x^3 + (16*n^2 + 32*n + 8)*x^4)*D[P[n - 1, x], x])/(20*n + (20*n^2 + 14*n)*x + (8*n^3 + 4*n^2)*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] = ((80*n - 20 + (100*n^2 + 126*n - 44)*x + (32*n^3 + 128*n^2 + 40*n - 28)*x^2 + (24*n^3 + 36*n^2 - 8*n - 4)*x^3)*v[i - 1] - (25 + (50*n + 80)*x + (20*n^2 + 130*n + 93)*x^2 + (36*n^2 + 112*n + 46)*x^3 + (16*n^2 + 32*n + 8)*x^4)*deriv(v[i - 1], x))/(20*n + (20*n^2 + 14*n)*x + (8*n^3 + 4*n^2)*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023