A143683 → γ → θ+1

nonnegative coefficients

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

Recurrence

\(\left(n + 8\,n^{2} x\right)\,P_{n}(x) = \left(-32\,x - 256\,x^{2} + n + 32\,n x + 128\,n x^{2} + 8\,n^{2} x + 128\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-x - 48\,x^{2} - 512\,x^{3} - 8\,n x^{2} - 256\,n x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	1	16							17
3	1	32							33
4	1	48	192						241
5	1	64	576						641
6	1	80	1152	2048					3281
7	1	96	1920	8192					10209
8	1	112	2880	20480	20480				43953
9	1	128	4032	40960	102400				147521
10	1	144	5376	71680	307200	196608			581009
11	1	160	6912	114688	716800	1179648			2018209
12	1	176	8640	172032	1433600	4128768	1835008		7578225
13	1	192	10560	245760	2580480	11010048	12845056		26692097
14	1	208	12672	337920	4300800	24772608	51380224	16777216	97581649

Technical details

Node id: n-778726377ce60171 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-9c2b720b33503d20

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: 1,  16;
  3: 1,  32;
  4: 1,  48,  192;
  5: 1,  64,  576;
  6: 1,  80, 1152,  2048;
  7: 1,  96, 1920,  8192;
  8: 1, 112, 2880, 20480,  20480;
  9: 1, 128, 4032, 40960, 102400;

%F Formula

The row polynomials satisfy (n+8*n^2*x)*P(n,x) = (n+(8*n^2+32*n-32)*x+(128*n^2+128*n-256)*x^2)*P(n-1,x) - (x+(8*n+48)*x^2+(256*n+512)*x^3)*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[((n + (8*n^2 + 32*n - 32)*x + (128*n^2 + 128*n - 256)*x^2)*P[n - 1, x] - (x + (8*n + 48)*x^2 + (256*n + 512)*x^3)*D[P[n - 1, x], x])/(n + 8*n^2*x)]];
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] = ((n + (8*n^2 + 32*n - 32)*x + (128*n^2 + 128*n - 256)*x^2)*v[i - 1] - (x + (8*n + 48)*x^2 + (256*n + 512)*x^3)*deriv(v[i - 1], x))/(n + 8*n^2*x)); 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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from gamma_polynomial of Pascal-(1,8,1) array.?
  - Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ? step ›
    
    from Pascal-(1,8,1) array.?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Pascal-(1,8,1) array.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Pascal-(1,8,1) array.?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.