basis-map coefficients of A371080, shifted → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 + 20\,x - 4\,x^{2} + 30\,n - 12\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-7 + 4\,x + 6\,n\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 1 + 4\,x\)

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	4														5
2	4	20	12													36
3	28	156	144	32												360
4	280	1672	1908	704	80											4644
5	3640	22856	29820	14240	2800	192										73548
6	58240	380256	545688	307360	80400	9792	448									1382184
7	1106560	7457824	11508840	7295008	2296000	379008	31360	1024								30075624
8	24344320	168498368	275567952	191180416	68749520	13848576	1574272	94208	2304							743859936
9	608608000	4309836480	7394693904	5514358272	2196689040	511213248	71670144	5953536	269568	5120						20613297312
10	17041024000	123109853440	219980938752	174121215360	75293188800	19586024640	3199594944	330516480	20943360	742400	11264					632684053440
11	528271744000	3884569552640	7188738661632	5984373512832	2769391891200	787870416960	144888167424	17559370752	1392906240	69555200	1982464	24576				21307127785920
12	17961239296000	134188451765760	256070823153408	222638669200640	109120258082880	33434134715520	6764561998656	928191559680	86867396352	5460224000	220425216	5160960	53248			781198883032320
13	664565853952000	5036817672517120	9877185811973376	8920486288832768	4594046222068160	1498951603873152	328301774953152	49804943705088	5302524674304	395066946560	20168225792	671883264	13152256	114688		30975878616871680
14	26582634158080000	204130970316492800	410197885496486400	383158613718573056	206063064604808320	70983775087889664	16629624740496768	2742601805239296	324162110308608	27586066027520	1675876314112	70878732288	1981837312	32915456	245760	1320842666874447360

Technical details

Node id: n-668b23a35a1d7e30 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-8b6922178d010988

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,          4;
  2:         4,         20,         12;
  3:        28,        156,        144,         32;
  4:       280,       1672,       1908,        704,         80;
  5:      3640,      22856,      29820,      14240,       2800,       192;
  6:     58240,     380256,     545688,     307360,      80400,      9792,      448;
  7:   1106560,    7457824,   11508840,    7295008,    2296000,    379008,    31360,    1024;
  8:  24344320,  168498368,  275567952,  191180416,   68749520,  13848576,  1574272,   94208,   2304;
  9: 608608000, 4309836480, 7394693904, 5514358272, 2196689040, 511213248, 71670144, 5953536, 269568, 5120;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-30*n+25+(12*n-20)*x+4*x^2)*P(n-2,x) + (6*n-7+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 30*n + 25 + (12*n - 20)*x + 4*x^2)*P[n - 2, x] + (6*n - 7 + 4*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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 30*n + 25 + (12*n - 20)*x + 4*x^2)*v[i - 2] + (6*n - 7 + 4*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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013609 of basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
    
    Parent-row provers of polytool polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
    
    ↑ Parent-row provers of polytool (above)

References

[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.