A038333 → 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

\(\left(47 + 28\,x + 28\,n\right)\,P_{n}(x) = \left(1425 + 2632\,x + 784\,x^{2} + 532\,n + 784\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1425 + 2632\,x + 784\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1425	2632	784															4841
1	37183	119700	110544	21952														289379
2	898529	4164496	6703200	4126976	614656													16507857
3	20721039	125794060	291514720	312816000	144444160	17210368												912500347
4	463030513	3481134552	10566701040	16324824320	13138272000	4853323776	481890304											49309176505
5	10114864415	90753980548	341151186096	690357801280	799916391680	515020262400	158541910016	13492928512										2619349324947
6	217210832577	2265729628960	10164445821376	25472621895168	38660036871680	35836254347264	19227423129600	5073341120512	377801998336									137294865645473
7	4602545584111	54737129809404	285481933248960	853813448995584	1604775179395584	1948465858332672	1505122682585088	692187232665600	159810245296128	10578455953408								7119574711866539
8	96483621635921	1288712763551080	7663198173316560	26644980436569600	59766941429690880	89867410046152704	90928406722191360	60204907303403520	24226553143296000	4971874298101760	296196766695424							365955664704604809
9	2004858666300927	29716955463863668	198461765586866320	786755012460500160	2051663493615859200	3681643592068958208	4613193715702505472	4000849895776419840	2317888931181035520	829086485348352000	153133728381534208	8293509467471872						18672691943719667395
10	41354041908867745	673632511877111472	4992448517929096224	22227717745729027840	66087421046682013440	137871786770985738240	206172041155861659648	221433298353720262656	168035695622609633280	86534520097425326080	27857305907704627200	4677539339654135808	232218265089212416					946836979377176712049
11	847699614002339663	15052871254827859180	122601117161634287904	605750420175397008512	2022722314861341533440	4811164252198450578432	8364221730773134786560	10720946140104806301696	10075215075094271950848	6796110356292211834880	3149856531546281869312	921823577309498572800	141885359969508786176	6502111422497947648				47754699557777865657051
12	17283776202987651249	332298248688917147896	2950362765946260399280	16019879309120213619456	59363541177188906834176	158581429485129176221696	314329397810298771124224	468396416923295548047360	525326360865135508783104	438831589937439400525824	266407525966654703927296	112249432760558408433664	30112903525443620044800	4278389316003649552384	182059119829942534144			2397378870986936014846553
13	350763935058686129119	7259186005254813524580	69782632224672601058160	413050787232476455899200	1682087327457622430042880	4986537458883868174070784	11100700063959042335518720	18859763868617926267453440	24590811888473016272486400	24515230173706323743211520	18430926777372454822084608	10171923718726815968133120	3928730146619544295178240	972878421591255416832000	128351679480109486571520	5097655355238390956032		119863482549640680159150323
14	7089586322951530805633	157142242906291385845312	1626057665177078229505920	10420873078884441758018560	46261688170037363060710400	150715024540202969731842048	372328130263328823663951872	710444804093378709473198080	1056146776642603870977392640	1224075969559545698897100800	1098282311782043303695876096	750641381478441796390354944	379751818832467796143636480	135390085052735064941527040	31132109490920173338624000	3833436827139269998936064	142734349946674946768896	5971357433656082278164094785

Technical details

Node id: n-d34ea40cdf30ed64 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-a55687d918dbb23b

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:             1425,              2632,                784;
  1:            37183,            119700,             110544,              21952;
  2:           898529,           4164496,            6703200,            4126976,              614656;
  3:         20721039,         125794060,          291514720,          312816000,           144444160,            17210368;
  4:        463030513,        3481134552,        10566701040,        16324824320,         13138272000,          4853323776,           481890304;
  5:      10114864415,       90753980548,       341151186096,       690357801280,        799916391680,        515020262400,        158541910016,         13492928512;
  6:     217210832577,     2265729628960,     10164445821376,     25472621895168,      38660036871680,      35836254347264,      19227423129600,       5073341120512,        377801998336;
  7:    4602545584111,    54737129809404,    285481933248960,    853813448995584,    1604775179395584,    1948465858332672,    1505122682585088,     692187232665600,     159810245296128,     10578455953408;
  8:   96483621635921,  1288712763551080,   7663198173316560,  26644980436569600,   59766941429690880,   89867410046152704,   90928406722191360,   60204907303403520,   24226553143296000,   4971874298101760,    296196766695424;
  9: 2004858666300927, 29716955463863668, 198461765586866320, 786755012460500160, 2051663493615859200, 3681643592068958208, 4613193715702505472, 4000849895776419840, 2317888931181035520, 829086485348352000, 153133728381534208, 8293509467471872;

%F Formula

The row polynomials satisfy (28*n+47+28*x)*P(n,x) = (532*n+1425+(784*n+2632)*x+784*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1425 + 2632*x + 784*x^2.

%t Mathematica

P[0, x_] := 1425 + 2632*x + 784*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((532*n + 1425 + (784*n + 2632)*x + 784*x^2)*P[n - 1, x])/(28*n + 47 + 28*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1425 + 2632*x + 784*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((532*n + 1425 + (784*n + 2632)*x + 784*x^2)*v[i - 1])/(28*n + 47 + 28*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)*12^(i-j)*7^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(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.