A059110 → D

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	4	2													6
4	21	18	3												42
5	136	156	48	4											344
6	1045	1460	630	100	5										3240
7	9276	15030	8160	1860	180	6									34512
8	93289	170142	109725	32060	4515	294	7								410032
9	1047376	2107448	1558368	549080	100240	9576	448	8							5372544
10	12975561	28393416	23508828	9616824	2137590	266616	18396	648	9						76917888
11	175721140	413689770	377055360	174531000	45417960	6941340	628320	32760	900	10					1194018560
12	2581284541	6483520010	6422902695	3302377320	979557810	176274252	19637310	1347720	54945	1210	11				19966957824
13	40864292184	108777654612	115975952400	65320138620	21661825680	4466229768	588196224	49836600	2680920	87780	1584	12			357706896384
14	693347907421	1945345308348	2214742136178	1351724262460	493894184355	114381733752	17323382076	1740240216	115836435	5013580	134706	2028	13		6832620141568
15	12548540320876	36941705020766	44623807064928	29259602976604	11645553499580	2984247988722	509418061152	59010127176	4667130468	250380130	8904896	199836	2548	14	138576811677696

Technical details

Node id: n-c0db4c865e1a9a1a · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-ced16c5d17801fb2

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
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:        0;
   2:        1;
   3:        4,        2;
   4:       21,       18,        3;
   5:      136,      156,       48,       4;
   6:     1045,     1460,      630,     100,       5;
   7:     9276,    15030,     8160,    1860,     180,      6;
   8:    93289,   170142,   109725,   32060,    4515,    294,     7;
   9:  1047376,  2107448,  1558368,  549080,  100240,   9576,   448,   8;
  10: 12975561, 28393416, 23508828, 9616824, 2137590, 266616, 18396, 648, 9;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n^2-2*n+1+(n-1)*x)*P(n-1,x) + (n-1+(n-1)*x)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - 2*n + 1 + (n - 1)*x)*P[n - 1, x] + (n - 1 + (n - 1)*x)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((n^2 - 2*n + 1 + (n - 1)*x)*v[i - 1] + (n - 1 + (n - 1)*x)*deriv(v[i - 1], x))/(n - 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 (−∞, 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n an…?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.