Real-rooted atlas

A204420 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-48 + 24\,x - 3\,x^{2} + 176\,n - 76\,n x + 8\,n x^{2} - 204\,n^{2} + 64\,n^{2} x - 4\,n^{2} x^{2} + 96\,n^{3} - 16\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(6 - 2\,x - 16\,n + 4\,n x + 8\,n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1022
2012921
3024027060570
401008013860504052529505
5072576011340005292009450056702489130
60798336001367150407484400017671500187110072765311008005
70124540416002288430144014045391360397296900056756700039729690108108053965081170

All 15 rows

Technical details
Node id
n-8f58599c2bca5578 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7811833956e55557

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,               2;
  2: 0,              12,                9;
  3: 0,             240,              270,               60;
  4: 0,           10080,            13860,             5040,             525;
  5: 0,          725760,          1134000,           529200,           94500,           5670;
  6: 0,        79833600,        136715040,         74844000,        17671500,        1871100,         72765;
  7: 0,     12454041600,      22884301440,      14045391360,      3972969000,      567567000,      39729690,      1081080;
  8: 0,   2615348736000,    5085919238400,    3407218214400,   1097674578000,   190702512000,   18275657400,    908107200,    18243225;
  9: 0, 711374856192000, 1450061425612800, 1042044190387200, 370970872272000, 74263644655200, 8753245300800, 602075073600, 22329707400, 344594250;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^4-96*n^3+204*n^2-176*n+48+(16*n^3-64*n^2+76*n-24)*x+(4*n^2-8*n+3)*x^2)*P(n-2,x) + (8*n^2-16*n+6+(4*n-2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 96*n^3 + 204*n^2 - 176*n + 48 + (16*n^3 - 64*n^2 + 76*n - 24)*x + (4*n^2 - 8*n + 3)*x^2)*P[n - 2, x] + (8*n^2 - 16*n + 6 + (4*n - 2)*x)*P[n - 1, x]];
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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 96*n^3 + 204*n^2 - 176*n + 48 + (16*n^3 - 64*n^2 + 76*n - 24)*x + (4*n^2 - 8*n + 3)*x^2)*v[i - 2] + (8*n^2 - 16*n + 6 + (4*n - 2)*x)*v[i - 1]); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.