A132393 → reverse → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0	1														1
1	1														1
2	2	2													4
3	6	14	8												28
4	24	92	116	48											280
5	120	652	1316	1168	384										3640
6	720	5112	14416	20168	13984	3840									58240
7	5040	44424	162256	314168	339904	194688	46080								1106560
8	40320	425952	1919984	4784928	7117584	6316160	3094272	645120							24344320
9	362880	4478688	24095088	73784096	140617104	170726784	128907008	55314432	10321920						608608000
10	3628800	51318720	321567264	1171552544	2734284768	4238375712	4362152192	2873470464	1098878976	185794560					17041024000
11	39916800	637081920	4563614304	19318423264	53508183328	101307828192	132751188352	118851218944	69557078016	24021319680	3715891200				528271744000
12	479001600	8523152640	68779173888	332220593856	1067103511744	2392873971520	3821786480448	4346740771072	3449411752960	1818511552512	573059727360	81749606400			17961239296000
13	6227020800	122297022720	1098684923904	5969567893440	21845639905216	56717845911616	107112199562304	148230505554688	149164131294208	106426532253696	51094053715968	14816178339840	1961990553600		664565853952000
14	87178291200	1874060858880	18561311525376	112139758529664	461047723902464	1362036480298240	2974234787574272	4860144266385536	5942290982540800	5368238865201152	3482406590619648	1535871893372928	412688504586240	51011754393600	26582634158080000

Technical details

Node id: n-bae2225dfab4d71f · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-2f2b2d9b62c79a93

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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:      1;
  2:      2,       2;
  3:      6,      14,        8;
  4:     24,      92,      116,       48;
  5:    120,     652,     1316,     1168,       384;
  6:    720,    5112,    14416,    20168,     13984,      3840;
  7:   5040,   44424,   162256,   314168,    339904,    194688,     46080;
  8:  40320,  425952,  1919984,  4784928,   7117584,   6316160,   3094272,   645120;
  9: 362880, 4478688, 24095088, 73784096, 140617104, 170726784, 128907008, 55314432, 10321920;

%F Formula

The row polynomials satisfy P(n,x) = (n+(2*n-2)*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[(n + (2*n - 2)*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] = (n + (2*n - 2)*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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from reversal of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.?
  - Affine, reversal and Möbius substitutions (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
  - Affine, reversal and Möbius substitutions (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 ?

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.