Real-rooted atlas

A100861 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(8 + 8\,x + 4\,n x + n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(-12\,x - 16\,x^{2} - 2\,x^{3} + 8\,n + 18\,n x + 8\,n x^{2} - n x^{3} + 6\,n^{2} x + 9\,n^{2} x^{2} + 2\,n^{2} x^{3} + n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-8 - 32\,x - 44\,x^{2} - 24\,x^{3} - 4\,x^{4} - 8\,n x - 22\,n x^{2} - 20\,n x^{3} - 6\,n x^{4} - 2\,n^{2} x^{2} - 4\,n^{2} x^{3} - 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2224
34610
41024943
5268045151
67630027060706
7232109212604203004
876442566300336052515205
92620167042948420160472573693
10949668760143640126000472505670400816
1135696288200689040720720346500623702142526
1214015212534723403620421344025987507484407276512430639
135685045568576168597002388672017567550648648094594571883475
142390480255076648554946413765752011981970056756700132432301081080442005838
Technical details
Node id
n-b496ec7a3d054de3 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-283b3aa9058db909

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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,     2;
  3:    4,     6;
  4:   10,    24,     9;
  5:   26,    80,    45;
  6:   76,   300,   270,    60;
  7:  232,  1092,  1260,   420;
  8:  764,  4256,  6300,  3360,  525;
  9: 2620, 16704, 29484, 20160, 4725;
%F Formula
The row polynomials satisfy (8+(4*n+8)*x+(n^2+n)*x^2)*P(n,x) = (8*n+(6*n^2+18*n-12)*x+(n^3+9*n^2+8*n-16)*x^2+(n^3+2*n^2-n-2)*x^3)*P(n-1,x) - (8+(8*n+32)*x+(2*n^2+22*n+44)*x^2+(4*n^2+20*n+24)*x^3+(2*n^2+6*n+4)*x^4)*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[((8*n + (6*n^2 + 18*n - 12)*x + (n^3 + 9*n^2 + 8*n - 16)*x^2 + (n^3 + 2*n^2 - n - 2)*x^3)*P[n - 1, x] - (8 + (8*n + 32)*x + (2*n^2 + 22*n + 44)*x^2 + (4*n^2 + 20*n + 24)*x^3 + (2*n^2 + 6*n + 4)*x^4)*D[P[n - 1, x], x])/(8 + (4*n + 8)*x + (n^2 + n)*x^2)]];
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] = ((8*n + (6*n^2 + 18*n - 12)*x + (n^3 + 9*n^2 + 8*n - 16)*x^2 + (n^3 + 2*n^2 - n - 2)*x^3)*v[i - 1] - (8 + (8*n + 32)*x + (2*n^2 + 22*n + 44)*x^2 + (4*n^2 + 20*n + 24)*x^3 + (2*n^2 + 6*n + 4)*x^4)*deriv(v[i - 1], x))/(8 + (4*n + 8)*x + (n^2 + n)*x^2)); 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.