A159854 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(7 + 14\,x + 8\,x^{2} + 2\,n + 8\,n x\right)\,P_{n}(x) = \left(18 + 80\,x + 104\,x^{2} + 32\,x^{3} + 4\,n + 24\,n x + 32\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 9 + 40\,x + 16\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	9	40	16															65
1	22	148	256	64														490
2	52	464	1360	1408	256													3540
3	120	1328	5408	9792	7168	1024												24840
4	272	3584	18560	47616	61696	34816	4096											170640
5	608	9280	58112	192000	355840	357376	163840	16384										1153440
6	1344	23296	170752	688128	1653760	2387968	1953792	753664	65536									7698240
7	2944	57088	478720	2272256	6694912	12587008	14852096	10240000	3407872	262144								50855040
8	6400	137216	1294336	7061504	24600576	56885248	87621632	87293952	51970048	15204352	1048576							333123840
9	13824	324608	3399680	20938752	84099072	230522880	437649408	570163200	491126784	257163264	67108864	4194304						2166704640
10	29696	757760	8720384	59801600	271810560	860749824	1940520960	3120562176	3519283200	2668625920	1246756864	293601280	16777216					14007997440
11	63488	1748992	21929984	165625856	839843840	3014787072	7863533568	15035006976	20952121344	20819476480	14094958592	5943328768	1275068416	67108864				90094602240
12	135168	3997696	54231040	447086592	2500788224	10029105152	29689380864	65745715200	109034078208	133982846976	118942072832	72712454144	27934064640	5502926848	268435456			576847319040
13	286720	9060352	132186112	1180827648	7219576832	31976062976	105862135808	266304749568	511916900352	749132054528	822905667584	659982123008	367723020288	129721434112	23622320128	1073741824		3678762147840
14	606208	20381696	318177280	3060793344	20300693504	98378448896	360021229568	1013637447680	2215007748096	3759455338496	4920640012288	4885525299200	3572540375040	1828313890816	596195147776	100931731456	4294967296	23378642288640

Technical details

Node id: n-97d46cba861f0d9a · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-29ea18b7c1a99aad

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:     9,     40,      16;
  1:    22,    148,     256,       64;
  2:    52,    464,    1360,     1408,      256;
  3:   120,   1328,    5408,     9792,     7168,      1024;
  4:   272,   3584,   18560,    47616,    61696,     34816,      4096;
  5:   608,   9280,   58112,   192000,   355840,    357376,    163840,     16384;
  6:  1344,  23296,  170752,   688128,  1653760,   2387968,   1953792,    753664,     65536;
  7:  2944,  57088,  478720,  2272256,  6694912,  12587008,  14852096,  10240000,   3407872,    262144;
  8:  6400, 137216, 1294336,  7061504, 24600576,  56885248,  87621632,  87293952,  51970048,  15204352,  1048576;
  9: 13824, 324608, 3399680, 20938752, 84099072, 230522880, 437649408, 570163200, 491126784, 257163264, 67108864, 4194304;

%F Formula

The row polynomials satisfy (2*n+7+(8*n+14)*x+8*x^2)*P(n,x) = (4*n+18+(24*n+80)*x+(32*n+104)*x^2+32*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 40*x + 16*x^2.

%t Mathematica

P[0, x_] := 9 + 40*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 18 + (24*n + 80)*x + (32*n + 104)*x^2 + 32*x^3)*P[n - 1, x])/(2*n + 7 + (8*n + 14)*x + 8*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 9 + 40*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 18 + (24*n + 80)*x + (32*n + 104)*x^2 + 32*x^3)*v[i - 1])/(2*n + 7 + (8*n + 14)*x + 8*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_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.