A053117 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	4							6
3	3	16	12						31
4	4	40	72	32					148
5	5	80	252	256	80				673
6	6	140	672	1152	800	192			2962
7	7	224	1512	3840	4400	2304	448		12735
8	8	336	3024	10560	17600	14976	6272	1024	53800

All 15 rows

Technical details

Node id: n-370478b5725584c9 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-71b82974c24ec403

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:  2,   4;
   3:  3,  16,   12;
   4:  4,  40,   72,    32;
   5:  5,  80,  252,   256,     80;
   6:  6, 140,  672,  1152,    800,    192;
   7:  7, 224, 1512,  3840,   4400,   2304,    448;
   8:  8, 336, 3024, 10560,  17600,  14976,   6272,   1024;
   9:  9, 480, 5544, 25344,  57200,  69888,  47040,  16384,  2304;
  10: 10, 660, 9504, 54912, 160160, 262080, 250880, 139264, 41472, 5120;

%F Formula

The row polynomials satisfy -(n-1-(n^3-3*n^2+3*n-1)*x)*P(n,x) = -(n-(n^3-2*n^2-n)*x-(n^3-n^2)*x^2)*P(n-1,x) - (2*x-(2*n^2-2*n-1)*x^2-(n^2-n)*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[Together[(-(n - (n^3 - 2*n^2 - n)*x - (n^3 - n^2)*x^2)*P[n - 1, x] - (2*x - (2*n^2 - 2*n - 1)*x^2 - (n^2 - n)*x^3)*D[P[n - 1, x], x])/(-n + 1 + (n^3 - 3*n^2 + 3*n - 1)*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 - (n^3 - 2*n^2 - n)*x - (n^3 - n^2)*x^2)*v[i - 1] - (2*x - (2*n^2 - 2*n - 1)*x^2 - (n^2 - n)*x^3)*deriv(v[i - 1], x))/(-n + 1 + (n^3 - 3*n^2 + 3*n - 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

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 shift_x_plus_one of Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing or…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Related ?

Same fingerprint ?

chebyshev_u → x ↦ x+1 → θ+1

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023