Real-rooted atlas

A142460 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 5\,n\right)\,P_{n}(x) = \left(4 - 72\,x - 36\,x^{2} + 20\,n - 320\,n x - 115\,n x^{2} + 200\,n^{2} x + 300\,n^{2} x^{2} - 125\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(20\,x - 20\,x^{3} + 100\,n x - 25\,n x^{2} - 75\,n x^{3} - 125\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-8 + 32\,x - 15\,n - 80\,n x + 50\,n^{2} x\right)\,P_{n-1}(x) + \left(-20\,x + 20\,x^{2} + 25\,n x - 25\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
2124328
311662494420
4110285478205658568
51621084930113240155256219912
613731211298453623680188307511193676785856
712239341388346389614540112018175277669267837698244290816

All 15 rows

Technical details
Node id
n-3afbc2ff27997910 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2d3f16c9e7bf1867

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: 1,       2;
  2: 1,      24,          3;
  3: 1,     166,        249,           4;
  4: 1,    1028,       5478,        2056,            5;
  5: 1,    6210,      84930,      113240,        15525,            6;
  6: 1,   37312,    1129845,     3623680,      1883075,       111936,           7;
  7: 1,  223934,   13883463,    89614540,    112018175,     27766926,      783769,          8;
  8: 1, 1343676,  163131024,  1915126024,   4704763350,   2872689036,   380639056,    5374704,        9;
  9: 1, 8062138, 1866999768, 37384765376, 161041626130, 193249951356, 65423339408, 4978666048, 36279621, 10;
%F Formula
The row polynomials satisfy (5*n-4)*P(n,x) = (20*n+4+(200*n^2-320*n-72)*x-(125*n^3-300*n^2+115*n+36)*x^2)*P(n-2,x) + ((100*n+20)*x-(125*n^2+25*n)*x^2+(125*n^2-75*n-20)*x^3)*d/dx P(n-2,x) - (15*n+8-(50*n^2-80*n+32)*x)*P(n-1,x) + ((25*n-20)*x-(25*n-20)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((20*n + 4 + (200*n^2 - 320*n - 72)*x - (125*n^3 - 300*n^2 + 115*n + 36)*x^2)*P[n - 2, x] + ((100*n + 20)*x - (125*n^2 + 25*n)*x^2 + (125*n^2 - 75*n - 20)*x^3)*D[P[n - 2, x], x] - (15*n + 8 - (50*n^2 - 80*n + 32)*x)*P[n - 1, x] + ((25*n - 20)*x - (25*n - 20)*x^2)*D[P[n - 1, x], x])/(5*n - 4)]];
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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((20*n + 4 + (200*n^2 - 320*n - 72)*x - (125*n^3 - 300*n^2 + 115*n + 36)*x^2)*v[i - 2] + ((100*n + 20)*x - (125*n^2 + 25*n)*x^2 + (125*n^2 - 75*n - 20)*x^3)*deriv(v[i - 2], x) - (15*n + 8 - (50*n^2 - 80*n + 32)*x)*v[i - 1] + ((25*n - 20)*x - (25*n - 20)*x^2)*deriv(v[i - 1], x))/(5*n - 4)); 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.