A053124 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	4							6
3	3	16	16						35
4	4	40	96	64					204
5	5	80	336	512	256				1189
6	6	140	896	2304	2560	1024			6930
7	7	224	2016	7680	14080	12288	4096		40391
8	8	336	4032	21120	56320	79872	57344	16384	235416

All 15 rows

Technical details

Node id: n-076ee53a8fbf87f2 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-9d7293d396b320fd

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,    16;
   4:  4,  40,    96,     64;
   5:  5,  80,   336,    512,    256;
   6:  6, 140,   896,   2304,   2560,    1024;
   7:  7, 224,  2016,   7680,  14080,   12288,    4096;
   8:  8, 336,  4032,  21120,  56320,   79872,   57344,   16384;
   9:  9, 480,  7392,  50688, 183040,  372736,  430080,  262144,   65536;
  10: 10, 660, 12672, 109824, 512512, 1397760, 2293760, 2228224, 1179648, 262144;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (n+2*n*x)*P(n-1,x) + (2*x+2*x^2)*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*x)*P[n - 1, x] + (2*x + 2*x^2)*D[P[n - 1, x], x])/(n - 1)]];
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*x)*v[i - 1] + (2*x + 2*x^2)*deriv(v[i - 1], x))/(n - 1)); 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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x 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