A163936 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + x + n\right)\,P_{n}(x) = \left(15\,x^{2} - 18\,x^{3} + 3\,x^{4} - 31\,n x^{2} + 21\,n x^{3} - 2\,n x^{4} + 20\,n^{2} x^{2} - 6\,n^{2} x^{3} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x^{2} + 9\,x^{3} - 6\,x^{4} + x^{5} + 6\,n x^{2} - 9\,n x^{3} + 3\,n x^{4} - 2\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 11\,x - 8\,x^{2} + x^{3} + n - 13\,n x + 5\,n x^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 3\,x^{2} - x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	1	5	2													8
3	1	17	26	6												50
4	1	45	196	154	24											420
5	1	105	1036	2104	1044	120										4410
6	1	229	4470	19052	22940	8028	720									55440
7	1	481	17070	134090	322820	262044	69264	5040								810810
8	1	989	60344	803150	3415900	5355844	3173256	663696	40320							13513500
9	1	2009	202784	4317260	29890000	80405884	89624136	40897656	6999840	362880						252702450
10	1	4053	658570	21523488	229151160	976909304	1827691712	1536400200	561238272	80627040	3628800					5237832600
11	1	8145	2089186	101670654	1597092504	10204675544	29667249920	41140006704	27209494896	8191046016	1007441280	39916800				119160691650
12	1	16333	6520044	461676130	10372290256	95345247480	407838752272	865484862976	930978545040	500082808464	126893211264	13575738240	479001600			2951518670100
13	1	32713	20115444	2035736800	63856348156	818610339120	4948363744912	15233229184816	24766977903120	21380662091664	9560303466624	2081962465920	196287356160	6227020800		79058535806250
14	1	65477	61556702	8779827220	377270090948	6583075126724	54528501558160	234034112678288	545749271247440	704778124863312	501313733753760	190300426270848	36093807256320	3031488633600	87178291200	2276885831220000

Technical details

Node id: n-fc743cefdb386d8c · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-31e93cd5a554a770

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 0)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1,    1;
  2: 1,    5,      2;
  3: 1,   17,     26,       6;
  4: 1,   45,    196,     154,       24;
  5: 1,  105,   1036,    2104,     1044,      120;
  6: 1,  229,   4470,   19052,    22940,     8028,      720;
  7: 1,  481,  17070,  134090,   322820,   262044,    69264,     5040;
  8: 1,  989,  60344,  803150,  3415900,  5355844,  3173256,   663696,   40320;
  9: 1, 2009, 202784, 4317260, 29890000, 80405884, 89624136, 40897656, 6999840, 362880;

%F Formula

The row polynomials satisfy (n-2+x)*P(n,x) = -((4*n^3-20*n^2+31*n-15)*x^2+(6*n^2-21*n+18)*x^3+(2*n-3)*x^4)*P(n-2,x) - ((2*n^2-6*n+4)*x^2-(2*n^2-9*n+9)*x^3-(3*n-6)*x^4-x^5)*d/dx P(n-2,x) + (n-2+(4*n^2-13*n+11)*x+(5*n-8)*x^2+x^3)*P(n-1,x) + ((n-2)*x-(n-3)*x^2-x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n^3 - 20*n^2 + 31*n - 15)*x^2 + (6*n^2 - 21*n + 18)*x^3 + (2*n - 3)*x^4)*P[n - 2, x] - ((2*n^2 - 6*n + 4)*x^2 - (2*n^2 - 9*n + 9)*x^3 - (3*n - 6)*x^4 - x^5)*D[P[n - 2, x], x] + (n - 2 + (4*n^2 - 13*n + 11)*x + (5*n - 8)*x^2 + x^3)*P[n - 1, x] + ((n - 2)*x - (n - 3)*x^2 - x^3)*D[P[n - 1, x], x])/(n - 2 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-((4*n^3 - 20*n^2 + 31*n - 15)*x^2 + (6*n^2 - 21*n + 18)*x^3 + (2*n - 3)*x^4)*v[i - 2] - ((2*n^2 - 6*n + 4)*x^2 - (2*n^2 - 9*n + 9)*x^3 - (3*n - 6)*x^4 - x^5)*deriv(v[i - 2], x) + (n - 2 + (4*n^2 - 13*n + 11)*x + (5*n - 8)*x^2 + x^3)*v[i - 1] + ((n - 2)*x - (n - 3)*x^2 - x^3)*deriv(v[i - 1], x))/(n - 2 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).?
    
    ↑ Liu–Wang theorem (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

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.