A038321 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(23 + 14\,x + 14\,n\right)\,P_{n}(x) = \left(666 + 1288\,x + 392\,x^{2} + 252\,n + 392\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1332 + 2576\,x + 784\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1332	2576	784															4692
1	33048	111888	108192	21952														275080
2	758160	3701376	6265728	4039168	614656													15379088
3	16586208	106142400	259096320	292400640	141370880	17210368												832806816
4	351459648	2786482944	8915961600	14509393920	12280826880	4750061568	481890304											44076076864
5	7278615936	68886091008	273075328512	582509491200	710960302080	481408413696	155168677888	13492928512										2292779848832
6	148157247744	1630409969664	7715242192896	20389624528896	32620531507200	31851021533184	17972580777984	4965397692416	377801998336									117670767448320
7	2975389355520	37335626431488	205431656177664	648080344203264	1284546345320448	1644074787962880	1337742904393728	647012908007424	156410027311104	10578455953408								5974188445116928
8	59111068529664	833109019545600	5226987700408320	19173621243248640	45365624094228480	71934595337945088	76723490104934400	53509716175749120	22645451780259840	4866089738567680	296196766695424							300633993030112256
9	1163972315879424	18206209107136512	128298789010022400	536637403908587520	1476368835730145280	2794522444204474368	3692642560681181184	3375833564617113600	2060124072766341120	774977683146670080	149875563947884544	8293509467471872						15016944608902907904
10	22751017168048128	391094698135486464	3058643129998934016	14369464369122508800	45077541928321351680	99211985761065762816	156493256875450564608	177246842912696696832	141785009713918771200	76911298716610068480	26039250153728114688	4578017226044473344	232218265089212416					745417374767349993472
11	441909587704799232	8281370249169518592	71179235060658536448	371115366439870660608	1307621257590148300800	3281645052381794402304	6018860469504656277504	8137649357523429359616	8064731352527699705856	5734415948429603635200	2799571273284606492672	861662459632457613312	138866522523349024768	6502111422497947648				36802543686157646274560
12	8537415594925178880	173228558380281298944	1623148568837225644032	9300753381259382095872	36369305911107324739584	102517506595067626782720	214400810088943900950528	337056186292260751540224	398744818518648038621184	351263854465650920521728	224789105178440462499840	99766539920687795011584	28147640347993615368192	4187359756088678285312	182059119829942534144			1808530854118790871072768
13	164168255000951488512	3585714549868575129600	36377997259859072778240	227240799637211590164480	976579105032235120066560	3055021696533015278125056	7176225461654733874790400	12864048605336634057031680	17695449780343689455861760	18608091530870241802321920	14753081887557338661912576	8582856743176817659084800	3491828897224072825405440	909385303550562958049280	125620792682660348559360	5097655355238390956032		88510656139019180621725696
14	3143934527278495629312	73547378240426266853376	803200059170560829030400	5432447590805621534883840	25450969559367698098421760	87501487810888266757963776	228108286674465140766670848	459278429545902967986585600	720386721898851507193774080	880840166843774764025118720	833642500582986832744022016	600852789602335247321530368	320426651745267859272499200	120333795842798817367818240	29100329713618014657576960	3751874341455455743639552	142734349946674946768896	4316129077474403134008786944

Technical details

Node id: n-d23383cdeaf9b1af · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-8ed4d1dc77260f2b

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:             1332,              2576,                784;
  1:            33048,            111888,             108192,              21952;
  2:           758160,           3701376,            6265728,            4039168,              614656;
  3:         16586208,         106142400,          259096320,          292400640,           141370880,            17210368;
  4:        351459648,        2786482944,         8915961600,        14509393920,         12280826880,          4750061568,           481890304;
  5:       7278615936,       68886091008,       273075328512,       582509491200,        710960302080,        481408413696,        155168677888,         13492928512;
  6:     148157247744,     1630409969664,      7715242192896,     20389624528896,      32620531507200,      31851021533184,      17972580777984,       4965397692416,        377801998336;
  7:    2975389355520,    37335626431488,    205431656177664,    648080344203264,    1284546345320448,    1644074787962880,    1337742904393728,     647012908007424,     156410027311104,     10578455953408;
  8:   59111068529664,   833109019545600,   5226987700408320,  19173621243248640,   45365624094228480,   71934595337945088,   76723490104934400,   53509716175749120,   22645451780259840,   4866089738567680,    296196766695424;
  9: 1163972315879424, 18206209107136512, 128298789010022400, 536637403908587520, 1476368835730145280, 2794522444204474368, 3692642560681181184, 3375833564617113600, 2060124072766341120, 774977683146670080, 149875563947884544, 8293509467471872;

%F Formula

The row polynomials satisfy (14*n+23+14*x)*P(n,x) = (252*n+666+(392*n+1288)*x+392*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1332 + 2576*x + 784*x^2.

%t Mathematica

P[0, x_] := 1332 + 2576*x + 784*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((252*n + 666 + (392*n + 1288)*x + 392*x^2)*P[n - 1, x])/(14*n + 23 + 14*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1332 + 2576*x + 784*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((252*n + 666 + (392*n + 1288)*x + 392*x^2)*v[i - 1])/(14*n + 23 + 14*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 Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
    
    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
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    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

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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.