A159854 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(11 + 18\,x + 9\,x^{2} + 3\,n + 9\,n x\right)\,P_{n}(x) = \left(28 + 96\,x + 99\,x^{2} + 27\,x^{3} + 6\,n + 27\,n x + 27\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 7 + 24\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	7	24	9															40
1	17	87	117	27														248
2	40	270	603	486	81													1480
3	92	768	2367	3267	1863	243												8600
4	208	2064	8064	15660	15471	6804	729											49000
5	464	5328	25128	62640	87885	67311	24057	2187										275000
6	1024	13344	73584	223344	405000	442746	276291	83106	6561									1525000
7	2240	32640	205776	734832	1630692	2313360	2066715	1086939	282123	19683								8375000
8	4864	78336	555264	2277504	5969376	10396512	12083904	9115416	4139991	944784	59049							45625000
9	10496	185088	1456128	6739200	20349792	41967072	60011280	58996512	38480265	15372423	3129597	177147						246875000
10	22528	431616	3730176	19215360	65629440	156250944	265029408	321016608	273200040	156873510	55919403	10274526	531441					1328125000
11	48128	995328	9370368	53146368	202435200	546061824	1070825184	1540417824	1616919084	1212472800	621608823	199998963	33480783	1594323				7109375000
12	102400	2273280	23150592	143299584	601945344	1813363200	4033877760	6715944576	8379919152	7756361712	5196312000	2405656260	705222207	108413964	4782969			37890625000
13	217088	5148672	56383488	378114048	1735748352	5773337856	14357730816	27140914944	39225227184	43188281136	35735037624	21628940112	9126436293	2456851743	349156737	14348907		201171875000
14	458752	11575296	135622656	979292160	4875945984	17741348352	48757572864	103119464448	169330487040	216076509792	212788976400	159140361744	87823813896	34038796050	8470638099	1119214746	43046721	1064453125000

Technical details

Node id: n-da0e7ae95fa3683e · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-8846766da2386d9b

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:     7,     24,       9;
  1:    17,     87,     117,      27;
  2:    40,    270,     603,     486,       81;
  3:    92,    768,    2367,    3267,     1863,      243;
  4:   208,   2064,    8064,   15660,    15471,     6804,      729;
  5:   464,   5328,   25128,   62640,    87885,    67311,    24057,     2187;
  6:  1024,  13344,   73584,  223344,   405000,   442746,   276291,    83106,     6561;
  7:  2240,  32640,  205776,  734832,  1630692,  2313360,  2066715,  1086939,   282123,    19683;
  8:  4864,  78336,  555264, 2277504,  5969376, 10396512, 12083904,  9115416,  4139991,   944784,   59049;
  9: 10496, 185088, 1456128, 6739200, 20349792, 41967072, 60011280, 58996512, 38480265, 15372423, 3129597, 177147;

%F Formula

The row polynomials satisfy (3*n+11+(9*n+18)*x+9*x^2)*P(n,x) = (6*n+28+(27*n+96)*x+(27*n+99)*x^2+27*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 7 + 24*x + 9*x^2.

%t Mathematica

P[0, x_] := 7 + 24*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 28 + (27*n + 96)*x + (27*n + 99)*x^2 + 27*x^3)*P[n - 1, x])/(3*n + 11 + (9*n + 18)*x + 9*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 7 + 24*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 28 + (27*n + 96)*x + (27*n + 99)*x^2 + 27*x^3)*v[i - 1])/(3*n + 11 + (9*n + 18)*x + 9*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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] 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.