Real-rooted atlas

A281260 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-4\,n - 2\,n^{2} + 4\,n^{2} x + 4\,n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(2 - 4\,x + 2\,x^{2} - 10\,n x + 8\,n x^{2} - 2\,n^{2} - 13\,n^{2} x + 11\,n^{2} x^{2} + 2\,n^{3} x + 12\,n^{3} x^{2} + n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x + 4\,x^{2} - 2\,x^{3} - 4\,n x + 10\,n x^{2} - 6\,n x^{3} + 8\,n^{2} x^{2} - 8\,n^{2} x^{3} + 2\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1044
204913
304241644
404458025154
5047224020036552
604105560875420492013
704144112028002520784647436
8041892016735010584617413448127742
9042403360168003528032928134402160100104312
10042975280346509979213582888704267303300121394706
11043607920660002494804656964435202138404950048401441501304
1204429114401179755662801387386181209612741304719008651568641695735188
13045041601620020011891883699696634233661158243303300968968144144946419621989840
14045852184032532523423409018009196310402484553518404100787286518738722306851274022584579165
Technical details
Node id
n-7d7a2499dfefbc27 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f7f59881146e5ced

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: 0, 4;
  2: 0, 4,   9;
  3: 0, 4,  24,   16;
  4: 0, 4,  45,   80,    25;
  5: 0, 4,  72,  240,   200,    36;
  6: 0, 4, 105,  560,   875,   420,    49;
  7: 0, 4, 144, 1120,  2800,  2520,   784,    64;
  8: 0, 4, 189, 2016,  7350, 10584,  6174,  1344,   81;
  9: 0, 4, 240, 3360, 16800, 35280, 32928, 13440, 2160, 100;
%F Formula
The row polynomials satisfy -(2*n^2+4*n-(n^4+4*n^3+4*n^2)*x)*P(n,x) = -(2*n^2-2-(n^4+2*n^3-13*n^2-10*n-4)*x-(3*n^4+12*n^3+11*n^2+8*n+2)*x^2)*P(n-1,x) - ((4*n+2)*x-(2*n^3+8*n^2+10*n+4)*x^2+(2*n^3+8*n^2+6*n+2)*x^3)*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[(-(2*n^2 - 2 - (n^4 + 2*n^3 - 13*n^2 - 10*n - 4)*x - (3*n^4 + 12*n^3 + 11*n^2 + 8*n + 2)*x^2)*P[n - 1, x] - ((4*n + 2)*x - (2*n^3 + 8*n^2 + 10*n + 4)*x^2 + (2*n^3 + 8*n^2 + 6*n + 2)*x^3)*D[P[n - 1, x], x])/(-2*n^2 - 4*n + (n^4 + 4*n^3 + 4*n^2)*x)]];
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] = (-(2*n^2 - 2 - (n^4 + 2*n^3 - 13*n^2 - 10*n - 4)*x - (3*n^4 + 12*n^3 + 11*n^2 + 8*n + 2)*x^2)*v[i - 1] - ((4*n + 2)*x - (2*n^3 + 8*n^2 + 10*n + 4)*x^2 + (2*n^3 + 8*n^2 + 6*n + 2)*x^3)*deriv(v[i - 1], x))/(-2*n^2 - 4*n + (n^4 + 4*n^3 + 4*n^2)*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.