A142075 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-5 + 4\,x + 2\,n\right)\,P_{n}(x) = \left(6 - 47\,x + 142\,x^{2} - 144\,x^{3} + 32\,x^{4} - 4\,n + 38\,n x - 142\,n x^{2} + 116\,n x^{3} - 16\,n x^{4} - 8\,n^{2} x + 56\,n^{2} x^{2} - 24\,n^{2} x^{3} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(3\,x - 25\,x^{2} + 64\,x^{3} - 60\,x^{4} + 16\,x^{5} - 2\,n x + 20\,n x^{2} - 44\,n x^{3} + 24\,n x^{4} - 4\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-13 + 57\,x - 50\,x^{2} + 8\,x^{3} + 6\,n - 38\,n x + 20\,n x^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-5\,x + 14\,x^{2} - 8\,x^{3} + 2\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	3	2							5
3	9	16	4						29
4	23	110	68	8					209
5	53	580	888	272	16				1809
6	115	2530	8456	6064	1072	32			18269
7	241	9768	62748	95552	38256	4224	64		210853
8	495	34838	392028	1124568	936784	232224	16704	128	2737769

All 15 rows

Technical details

Node id: n-54f2950c75bd7fde · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-6093ad2f7ab921a1

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:    3,      2;
   3:    9,     16,        4;
   4:   23,    110,       68,        8;
   5:   53,    580,      888,      272,        16;
   6:  115,   2530,     8456,     6064,      1072,        32;
   7:  241,   9768,    62748,    95552,     38256,      4224,       64;
   8:  495,  34838,   392028,  1124568,    936784,    232224,    16704,     128;
   9: 1005, 117868,  2176048, 10702032,  16616480,   8432960,  1384704,   66304,    256;
  10: 2027, 384746, 11115968, 87504192, 230622304, 216725056, 71996928, 8198144, 263936, 512;

%F Formula

The row polynomials satisfy (2*n-5+4*x)*P(n,x) = -(4*n-6+(8*n^2-38*n+47)*x+(8*n^3-56*n^2+142*n-142)*x^2+(24*n^2-116*n+144)*x^3+(16*n-32)*x^4)*P(n-2,x) - ((2*n-3)*x+(4*n^2-20*n+25)*x^2-(8*n^2-44*n+64)*x^3-(24*n-60)*x^4-16*x^5)*d/dx P(n-2,x) + (6*n-13+(8*n^2-38*n+57)*x+(20*n-50)*x^2+8*x^3)*P(n-1,x) + ((2*n-5)*x-(4*n-14)*x^2-8*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 6 + (8*n^2 - 38*n + 47)*x + (8*n^3 - 56*n^2 + 142*n - 142)*x^2 + (24*n^2 - 116*n + 144)*x^3 + (16*n - 32)*x^4)*P[n - 2, x] - ((2*n - 3)*x + (4*n^2 - 20*n + 25)*x^2 - (8*n^2 - 44*n + 64)*x^3 - (24*n - 60)*x^4 - 16*x^5)*D[P[n - 2, x], x] + (6*n - 13 + (8*n^2 - 38*n + 57)*x + (20*n - 50)*x^2 + 8*x^3)*P[n - 1, x] + ((2*n - 5)*x - (4*n - 14)*x^2 - 8*x^3)*D[P[n - 1, x], x])/(2*n - 5 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(4*n - 6 + (8*n^2 - 38*n + 47)*x + (8*n^3 - 56*n^2 + 142*n - 142)*x^2 + (24*n^2 - 116*n + 144)*x^3 + (16*n - 32)*x^4)*v[i - 2] - ((2*n - 3)*x + (4*n^2 - 20*n + 25)*x^2 - (8*n^2 - 44*n + 64)*x^3 - (24*n - 60)*x^4 - 16*x^5)*deriv(v[i - 2], x) + (6*n - 13 + (8*n^2 - 38*n + 57)*x + (20*n - 50)*x^2 + 8*x^3)*v[i - 1] + ((2*n - 5)*x - (4*n - 14)*x^2 - 8*x^3)*deriv(v[i - 1], x))/(2*n - 5 + 4*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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Related ?

Same fingerprint ?

A156365 → D+1

References

[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.