A225117 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-11 + 6\,x + 3\,n\right)\,P_{n}(x) = \left(80 - 460\,x + 1244\,x^{2} - 900\,x^{3} + 144\,x^{4} - 30\,n + 246\,n x - 954\,n x^{2} + 540\,n x^{3} - 54\,n x^{4} - 36\,n^{2} x + 270\,n^{2} x^{2} - 81\,n^{2} x^{3} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(48\,x - 348\,x^{2} + 570\,x^{3} - 324\,x^{4} + 54\,x^{5} - 18\,n x + 189\,n x^{2} - 252\,n x^{3} + 81\,n x^{4} - 27\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-71 + 251\,x - 156\,x^{2} + 18\,x^{3} + 21\,n - 123\,n x + 45\,n x^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-33\,x + 51\,x^{2} - 18\,x^{3} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	3	1							4
3	17	15	1						33
4	101	213	63	1					378
5	561	2807	1884	255	1				5508
6	2965	31929	46532	14750	1023	1			97200
7	15241	323403	941427	618430	109443	4095	1		2012040
8	77229	3028573	16146597	20371695	7332847	790755	16383	1	47764080

All 15 rows

Technical details

Node id: n-c18c59cf0567612d · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-42411ce3f1f92da1

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,         1;
   3:      17,        15,          1;
   4:     101,       213,         63,           1;
   5:     561,      2807,       1884,         255,           1;
   6:    2965,     31929,      46532,       14750,        1023,          1;
   7:   15241,    323403,     941427,      618430,      109443,       4095,         1;
   8:   77229,   3028573,   16146597,    20371695,     7332847,     790755,     16383,        1;
   9:  388577,  26929935,  245485606,   548439569,   370138674,   81526081,   5633702,    65535,      1;
  10: 1948517, 231367041, 3425508306, 12652000212, 14831873244, 6038516358, 872385774, 39829284, 262143, 1;

%F Formula

The row polynomials satisfy (3*n-11+6*x)*P(n,x) = -(30*n-80+(36*n^2-246*n+460)*x+(27*n^3-270*n^2+954*n-1244)*x^2+(81*n^2-540*n+900)*x^3+(54*n-144)*x^4)*P(n-2,x) - ((18*n-48)*x+(27*n^2-189*n+348)*x^2-(27*n^2-252*n+570)*x^3-(81*n-324)*x^4-54*x^5)*d/dx P(n-2,x) + (21*n-71+(18*n^2-123*n+251)*x+(45*n-156)*x^2+18*x^3)*P(n-1,x) + ((9*n-33)*x-(9*n-51)*x^2-18*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(30*n - 80 + (36*n^2 - 246*n + 460)*x + (27*n^3 - 270*n^2 + 954*n - 1244)*x^2 + (81*n^2 - 540*n + 900)*x^3 + (54*n - 144)*x^4)*P[n - 2, x] - ((18*n - 48)*x + (27*n^2 - 189*n + 348)*x^2 - (27*n^2 - 252*n + 570)*x^3 - (81*n - 324)*x^4 - 54*x^5)*D[P[n - 2, x], x] + (21*n - 71 + (18*n^2 - 123*n + 251)*x + (45*n - 156)*x^2 + 18*x^3)*P[n - 1, x] + ((9*n - 33)*x - (9*n - 51)*x^2 - 18*x^3)*D[P[n - 1, x], x])/(3*n - 11 + 6*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 + x; for(n = 3, N, my(i = n); v[i] = (-(30*n - 80 + (36*n^2 - 246*n + 460)*x + (27*n^3 - 270*n^2 + 954*n - 1244)*x^2 + (81*n^2 - 540*n + 900)*x^3 + (54*n - 144)*x^4)*v[i - 2] - ((18*n - 48)*x + (27*n^2 - 189*n + 348)*x^2 - (27*n^2 - 252*n + 570)*x^3 - (81*n - 324)*x^4 - 54*x^5)*deriv(v[i - 2], x) + (21*n - 71 + (18*n^2 - 123*n + 251)*x + (45*n - 156)*x^2 + 18*x^3)*v[i - 1] + ((9*n - 33)*x - (9*n - 51)*x^2 - 18*x^3)*deriv(v[i - 1], x))/(3*n - 11 + 6*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 read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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.