Real-rooted atlas

A089503 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-1 + 2\,x^{2} + n - 2\,n x - 3\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} + x^{3} - n - 3\,n x - 3\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2549
3437230145
4529136811763363409
585012992039420230405040105921
616902175510013477201201200534600950404102681
7401045521758184491427305914272039999960144144002162160190630609
811067683370728011219358186402941528800267998640014639788804439635205765760010340890785

All 15 rows

Technical details
Node id
n-f573c9322e5df2c8 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-11ac09149027f91e

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:            1;
   2:            5,             4;
   3:           43,            72,            30;
   4:          529,          1368,          1176,           336;
   5:         8501,         29920,         39420,         23040,           5040;
   6:       169021,        755100,       1347720,       1201200,         534600,          95040;
   7:      4010455,      21758184,      49142730,      59142720,       39999960,       14414400,       2162160;
   8:    110676833,     707280112,    1935818640,    2941528800,     2679986400,     1463978880,     443963520,      57657600;
   9:   3484717129,   25651375488,   82568248056,  151793703936,   174319387200,   128051159040,   58757287680,   15397724160,   1764322560;
  10: 123320412181, 1027994161140, 3807078240240, 8221185413280, 11408087310480, 10549156020480, 6500479910400, 2573936870400, 594255916800, 60949324800;
%F Formula
The row polynomials satisfy P(n,x) = (n^2+n-1+(2*n^2-2*n)*x+(n^2-3*n+2)*x^2)*P(n-1,x) - (n+(3*n-1)*x+(3*n-2)*x^2+(n-1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n^2 + n - 1 + (2*n^2 - 2*n)*x + (n^2 - 3*n + 2)*x^2)*P[n - 1, x] - (n + (3*n - 1)*x + (3*n - 2)*x^2 + (n - 1)*x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (n^2 + n - 1 + (2*n^2 - 2*n)*x + (n^2 - 3*n + 2)*x^2)*v[i - 1] - (n + (3*n - 1)*x + (3*n - 2)*x^2 + (n - 1)*x^3)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.