A132393 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3 + n x\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	3	1														4
2	9	9	2													20
3	27	54	33	6												120
4	81	270	315	150	24											840
5	243	1215	2295	2025	822	120										6720
6	729	5103	14175	19845	14616	5292	720									60480
7	2187	20412	78246	158760	182763	118188	39204	5040								604800
8	6561	78732	398034	1102248	1818369	1816668	1063116	328752	40320							6652800
9	19683	295245	1902690	6889050	15375339	21815325	19539360	10554300	3079728	362880						79833600
10	59049	1082565	8660520	39694050	115016517	219199365	276771330	227056500	114782184	31885920	3628800					1037836800
11	177147	3897234	37889775	214347870	781684101	1922779782	3241507005	3725654130	2841968052	1358261784	361631520	39916800				14529715200
12	531441	13817466	160436133	1097720910	4917226743	15148548558	32797878399	50075046450	53233753716	38178401976	17384035968	4459328640	479001600			217945728000
13	1594323	48361131	660935457	5378832459	29022052059	109369593333	295324766451	576597558537	810676864998	806574004236	548471333592	239370453504	59408277120	6227020800		3487131648000
14	4782969	167403915	2659862205	25389593775	162369810603	734417508825	2417148606015	5864339405925	10504396414512	13769198122680	12937450060080	8396710030800	3529411180416	850396942080	87178291200	59281238016000

Technical details

Node id: n-262f1128a1842f2b · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-2a63385a6c1d1a3f

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:     1;
  1:     3,      1;
  2:     9,      9,       2;
  3:    27,     54,      33,       6;
  4:    81,    270,     315,     150,       24;
  5:   243,   1215,    2295,    2025,      822,      120;
  6:   729,   5103,   14175,   19845,    14616,     5292,      720;
  7:  2187,  20412,   78246,  158760,   182763,   118188,    39204,     5040;
  8:  6561,  78732,  398034, 1102248,  1818369,  1816668,  1063116,   328752,   40320;
  9: 19683, 295245, 1902690, 6889050, 15375339, 21815325, 19539360, 10554300, 3079728, 362880;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3 + n*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3 + n*x)*v[i - 1]); 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A013610 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 …?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A132393 → basis A013610

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.