A159854 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 18\,x + 18\,x^{2} + 3\,n x + 9\,n x^{2}\right)\,P_{n}(x) = \left(4 + 48\,x + 117\,x^{2} + 81\,x^{3} + 6\,n x + 27\,n x^{2} + 27\,n x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1 + 12\,x + 27\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	12	27															40
1	2	30	108	108														248
2	4	72	351	648	405													1480
3	8	168	1026	2700	3240	1458												8600
4	16	384	2808	9504	16605	14580	5103											49000
5	32	864	7344	30240	68850	88938	61236	17496										275000
6	64	1920	18576	89856	251100	425736	433755	244944	59049									1525000
7	128	4224	45792	254016	839160	1755432	2357586	1977048	944784	196830								8375000
8	256	9216	110592	691200	2630880	6531840	10859184	12037248	8562105	3542940	649539							45625000
9	512	19968	262656	1824768	7853760	22534848	44579808	61305984	57749922	35626230	12990780	2125764						246875000
10	1024	43008	615168	4700160	22550400	73343232	168031584	275457024	322407540	263752200	143548119	46766808	6908733					1328125000
11	2048	92160	1423872	11860992	62726400	227867904	592764480	1127022336	1574482536	1602196200	1157478498	563327460	165809592	22320522				7109375000
12	4096	196608	3262464	29417472	169931520	681550848	1983066624	4286380032	6952665456	8452667520	7602204456	4914766368	2162433429	580333572	71744535			37890625000
13	8192	417792	7409664	71884800	450178560	1974855168	6351561216	15371845632	28371863520	40080886560	43098211728	34709474592	20297857554	8146990530	2008846980	229582512		201171875000
14	16384	884736	16699392	173408256	1169925120	5570353152	19615115520	52520472576	108623706048	174734651520	218463349104	210467642112	153401507532	81871674696	30204449235	6887475360	731794257	1064453125000

Technical details

Node id: n-8d161f6025855749 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-9e4b77a2d9d784c8

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   1,    12,     27;
  1:   2,    30,    108,     108;
  2:   4,    72,    351,     648,     405;
  3:   8,   168,   1026,    2700,    3240,     1458;
  4:  16,   384,   2808,    9504,   16605,    14580,     5103;
  5:  32,   864,   7344,   30240,   68850,    88938,    61236,    17496;
  6:  64,  1920,  18576,   89856,  251100,   425736,   433755,   244944,    59049;
  7: 128,  4224,  45792,  254016,  839160,  1755432,  2357586,  1977048,   944784,   196830;
  8: 256,  9216, 110592,  691200, 2630880,  6531840, 10859184, 12037248,  8562105,  3542940,   649539;
  9: 512, 19968, 262656, 1824768, 7853760, 22534848, 44579808, 61305984, 57749922, 35626230, 12990780, 2125764;

%F Formula

The row polynomials satisfy (2+(3*n+18)*x+(9*n+18)*x^2)*P(n,x) = (4+(6*n+48)*x+(27*n+117)*x^2+(27*n+81)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 12*x + 27*x^2.

%t Mathematica

P[0, x_] := 1 + 12*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4 + (6*n + 48)*x + (27*n + 117)*x^2 + (27*n + 81)*x^3)*P[n - 1, x])/(2 + (3*n + 18)*x + (9*n + 18)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 12*x + 27*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4 + (6*n + 48)*x + (27*n + 117)*x^2 + (27*n + 81)*x^3)*v[i - 1])/(2 + (3*n + 18)*x + (9*n + 18)*x^2)); 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_A013610 of Riordan array (1-x,x/(1-x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    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
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

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.