A053117 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0)coefficients of one sign Hadamard kernel

Recurrence

\(\left(-1 - 2\,x - x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(4\,x + 14\,x^{2} + 16\,x^{3} + 6\,x^{4} + n + 2\,n x - 2\,n x^{2} - 6\,n x^{3} - 3\,n x^{4}\right)\,P_{n-1}(x) + \left(2\,x + 9\,x^{2} + 15\,x^{3} + 11\,x^{4} + 3\,x^{5}\right)\,P_{n-1}'(x)\)

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

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	14	15													32
4	4	32	76	56												168
5	5	60	234	372	209											880
6	6	100	564	1440	1718	780										4608
7	7	154	1169	4236	8009	7642	2911									24128
8	8	224	2184	10480	27800	41664	33112	10864								126336
9	9	312	3780	22968	79870	166632	206676	140712	40545							661504
10	10	420	6168	45984	200668	545000	935640	989376	589098	151316						3463680
11	11	550	9603	85800	455686	1543332	3431470	5000616	4607055	2437222	564719					18136064
12	12	704	14388	151272	955768	3917088	10809960	20321456	25713084	20983776	9986596	2107560				94961664
13	13	884	20878	254540	1879735	9116744	30314452	70475864	114672235	128167940	93862078	40594844	7865521			497225728
14	14	1092	29484	411840	3504930	19772532	77495976	216351072	434081970	622293420	622728684	413573472	163908702	29354524		2603507712
15	15	1330	40677	644436	6247527	40434030	183654405	602464536	1447629813	2552926446	3269760855	2961574740	1799194005	658017970	109552575	13632143360

Technical details

Node id: n-de6b5f28ca03e00c · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-0878106a64e78b92

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,  14,   15;
   4:  4,  32,   76,    56;
   5:  5,  60,  234,   372,    209;
   6:  6, 100,  564,  1440,   1718,    780;
   7:  7, 154, 1169,  4236,   8009,   7642,   2911;
   8:  8, 224, 2184, 10480,  27800,  41664,  33112,  10864;
   9:  9, 312, 3780, 22968,  79870, 166632, 206676, 140712,  40545;
  10: 10, 420, 6168, 45984, 200668, 545000, 935640, 989376, 589098, 151316;

%F Formula

The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = (n+(2*n+4)*x-(2*n-14)*x^2-(6*n-16)*x^3-(3*n-6)*x^4)*P(n-1,x) + (2*x+9*x^2+15*x^3+11*x^4+3*x^5)*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 + (2*n + 4)*x - (2*n - 14)*x^2 - (6*n - 16)*x^3 - (3*n - 6)*x^4)*P[n - 1, x] + (2*x + 9*x^2 + 15*x^3 + 11*x^4 + 3*x^5)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)]];
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 + (2*n + 4)*x - (2*n - 14)*x^2 - (6*n - 16)*x^3 - (3*n - 6)*x^4)*v[i - 1] + (2*x + 9*x^2 + 15*x^3 + 11*x^4 + 3*x^5)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x + (n - 1)*x^2)); 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 [-1, 0)
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) 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 ?

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023