A159854 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + 4\,x + n + 4\,n x\right)\,P_{n}(x) = \left(8 + 32\,x + 32\,x^{2} + 2\,n + 12\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 8 + 32\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	8	32															40
1	20	128	192														340
2	48	416	1152	1024													2640
3	112	1216	4800	8192	5120												19440
4	256	3328	16896	41984	51200	24576											138240
5	576	8704	53760	174080	312320	294912	114688										959040
6	1280	22016	159744	634880	1495040	2088960	1605632	524288									6531840
7	2816	54272	451584	2121728	6164480	11354112	12959744	8388608	2359296								43856640
8	6144	131072	1228800	6651904	22937600	52297728	78905344	76021760	42467328	10485760							291133440
9	13312	311296	3244032	19857408	79134720	214695936	401866752	512753664	427032576	209715200	46137344						1914762240
10	28672	729088	8355840	57016320	257556480	809238528	1805647872	2862612480	3161456640	2317352960	1015021568	201326592					12496343040
11	61440	1687552	21086208	158597120	800194560	2854748160	7387742208	13979615232	19204669440	18685624320	12226396160	4831838208	872415232				81024675840
12	131072	3866624	52297728	429654016	2393374720	9550430208	28097642496	61731766272	101317607424	122725335040	106669539328	63015223296	22682796032	3758096384			522427760640
13	278528	8781824	127795200	1138229248	6935019520	30589059072	100763959296	251909898240	480428163072	695751147520	753330552832	591497527296	318431559680	105226698752	16106127360		3352244797440
14	589824	19791872	308281344	2958032896	19561185280	94466211840	344276860928	964455038976	2094450868224	3526738575360	4567966154752	4470255648768	3200019070976	1582158577664	483183820800	68719476736	21419538186240

Technical details

Node id: n-db2d48c86b57e427 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-657752d90232a0f4

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
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:     8,     32;
  1:    20,    128,     192;
  2:    48,    416,    1152,     1024;
  3:   112,   1216,    4800,     8192,     5120;
  4:   256,   3328,   16896,    41984,    51200,     24576;
  5:   576,   8704,   53760,   174080,   312320,    294912,    114688;
  6:  1280,  22016,  159744,   634880,  1495040,   2088960,   1605632,    524288;
  7:  2816,  54272,  451584,  2121728,  6164480,  11354112,  12959744,   8388608,   2359296;
  8:  6144, 131072, 1228800,  6651904, 22937600,  52297728,  78905344,  76021760,  42467328,  10485760;
  9: 13312, 311296, 3244032, 19857408, 79134720, 214695936, 401866752, 512753664, 427032576, 209715200, 46137344;

%F Formula

The row polynomials satisfy (n+3+(4*n+4)*x)*P(n,x) = (2*n+8+(12*n+32)*x+(16*n+32)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 8 + 32*x.

%t Mathematica

P[0, x_] := 8 + 32*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 8 + (12*n + 32)*x + (16*n + 32)*x^2)*P[n - 1, x])/(n + 3 + (4*n + 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 8 + 32*x; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 8 + (12*n + 32)*x + (16*n + 32)*x^2)*v[i - 1])/(n + 3 + (4*n + 4)*x)); 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_A013611 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.