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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
9	9	480	5544	25344	57200	69888	47040	16384	2304							224193
10	10	660	9504	54912	160160	262080	250880	139264	41472	5120						924062
11	11	880	15444	109824	400400	838656	1066240	835584	393984	102400	11264					3774687
12	12	1144	24024	205920	915200	2376192	3838464	3969024	2626560	1075200	247808	24576				15304124
13	13	1456	36036	366080	1944800	6110208	12155136	15876096	13789440	7884800	2849792	589824	53248			61656929
14	14	1820	52416	622336	3889600	14511744	34728960	55566336	60673536	45337600	22798336	7372800	1384448	114688		247054634
15	15	2240	74256	1018368	7390240	32248320	91163520	174637056	232581888	217620480	142489600	63897600	18690048	3211264	245760	985270655

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