Real-rooted atlas

A257606 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(3 + n\right)\,P_{n}(x) = \left(-48 - 48\,x - 24\,x^{2} - 12\,n - 36\,n x - 26\,n x^{2} - 6\,n^{2} x - 9\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-12\,x + 12\,x^{3} - 3\,n x - 4\,n x^{2} + 7\,n x^{3} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(24 + 18\,x + 7\,n + 12\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
2168048144
3645928882561800
42563856106567712128023760
5102423376104424139232584406144332640
6409613531290705619492481511760405936286724942080
7163847584807269048233200002915000014538096265468013107277837760

All 15 rows

Technical details
Node id
n-b15ed6e1650924b8 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3dd8844cb1c111f1

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:      4,        8;
  2:     16,       80,        48;
  3:     64,      592,       888,        256;
  4:    256,     3856,     10656,       7712,       1280;
  5:   1024,    23376,    104424,     139232,      58440,       6144;
  6:   4096,   135312,    907056,    1949248,    1511760,     405936,      28672;
  7:  16384,   758480,   7269048,   23320000,   29150000,   14538096,    2654680,     131072;
  8:  65536,  4152848,  54991488,  250468576,  466400000,  375702864,  128313472,   16611392,    589824;
  9: 262144, 22337104, 398470920, 2486499872, 6548971480, 7858765776, 4351374776, 1062589120, 100516968, 2621440;
%F Formula
The row polynomials satisfy (n+3)*P(n,x) = -(12*n+48+(6*n^2+36*n+48)*x+(n^3+9*n^2+26*n+24)*x^2)*P(n-2,x) - ((3*n+12)*x+(n^2+4*n)*x^2-(n^2+7*n+12)*x^3)*d/dx P(n-2,x) + (7*n+24+(2*n^2+12*n+18)*x)*P(n-1,x) + ((n+3)*x-(n+3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n + 48 + (6*n^2 + 36*n + 48)*x + (n^3 + 9*n^2 + 26*n + 24)*x^2)*P[n - 2, x] - ((3*n + 12)*x + (n^2 + 4*n)*x^2 - (n^2 + 7*n + 12)*x^3)*D[P[n - 2, x], x] + (7*n + 24 + (2*n^2 + 12*n + 18)*x)*P[n - 1, x] + ((n + 3)*x - (n + 3)*x^2)*D[P[n - 1, x], x])/(n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n + 48 + (6*n^2 + 36*n + 48)*x + (n^3 + 9*n^2 + 26*n + 24)*x^2)*v[i - 2] - ((3*n + 12)*x + (n^2 + 4*n)*x^2 - (n^2 + 7*n + 12)*x^3)*deriv(v[i - 2], x) + (7*n + 24 + (2*n^2 + 12*n + 18)*x)*v[i - 1] + ((n + 3)*x - (n + 3)*x^2)*deriv(v[i - 1], x))/(n + 3)); 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.