A132393 → θ+1 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 + 4\,x - 4\,x^{2} - 2\,n x + 4\,n x^{2} - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(2 - 3\,x + 2\,n x\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0	1														1
1	2														2
2	3	2													5
3	4	9	4												17
4	5	24	33	12											74
5	6	50	140	150	48										394
6	7	90	425	900	822	240									2484
7	8	147	1050	3675	6496	5292	1440								18108
8	9	224	2254	11760	33845	52528	39204	10080							149904
9	10	324	4368	31752	134694	336420	472496	328752	80640						1389456
10	11	450	7830	75600	442911	1615950	3618400	4690800	3079728	725760					14257440
11	12	605	13200	163350	1262184	6314385	20501580	42047500	51014304	31885920	7257600				160460640
12	13	792	21175	326700	3216807	21100464	93376745	275974380	526290380	603671904	361631520	79833600			1965444480
13	14	1014	32604	613470	7494630	62339706	359921848	1442491050	3943241016	7070074440	7726238208	4459328640	958003200		26029779840
14	15	1274	48503	1093092	16219203	166696530	1215328257	6327545224	23352831502	59746222536	101568765480	106386868224	59408277120	12454041600	370643938560

Technical details

Node id: n-bf568154c856fd5d · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-269f20a8987741e2

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:  1;
  1:  2;
  2:  3,   2;
  3:  4,   9,    4;
  4:  5,  24,   33,    12;
  5:  6,  50,  140,   150,     48;
  6:  7,  90,  425,   900,    822,    240;
  7:  8, 147, 1050,  3675,   6496,   5292,   1440;
  8:  9, 224, 2254, 11760,  33845,  52528,  39204,  10080;
  9: 10, 324, 4368, 31752, 134694, 336420, 472496, 328752, 80640;

%F Formula

The row polynomials satisfy P(n,x) = -(1+(2*n-4)*x+(n^2-4*n+4)*x^2)*P(n-2,x) + (2+(2*n-3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(1 + (2*n - 4)*x + (n^2 - 4*n + 4)*x^2)*P[n - 2, x] + (2 + (2*n - 3)*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; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = -(1 + (2*n - 4)*x + (n^2 - 4*n + 4)*x^2)*v[i - 2] + (2 + (2*n - 3)*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 theta_plus_one of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= …?
  - Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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
  - Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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 → θ+1

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.