A038321 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(13 + 7\,x + 7\,n\right)\,P_{n}(x) = \left(360 + 546\,x + 147\,x^{2} + 126\,n + 147\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1080 + 1638\,x + 441\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1080	1638	441															3159
1	26244	68040	51597	9261														155142
2	594864	2204496	2857680	1444716	194481													7296237
3	12912048	62460720	115736040	100018800	37923795	4084101												333135504
4	272097792	1626918048	3935025360	4860913680	3150592200	955679634	85766121											14886992835
5	5612016960	39998375424	119578476528	192816242640	178638577740	92627410680	23414151033	1801088541										654486339546
6	113872925952	942818849280	3359863535616	6696394685568	8098282190880	6002256212064	2593567499040	561939624792	37822859361									28406818382553
7	2281131839232	21521983004928	89096381256960	211671402743808	316404648893088	306115066815264	189071070680016	70026322474080	13275823635711	794280046581								1220258111389668
8	45225918203904	479037686238720	2259808215517440	6236746687987200	11112748644049920	13288995253509696	10714027338534240	5672132120400480	1838190964944600	309769218166590	16679880978201							51973361928530991
9	889046339429376	10447187105101824	55328852760572160	174005232594842880	360172121231260800	513408987355106304	511626317260123296	353562902171629920	163782814976563860	47180234766911400	7155668939648229	350277500542221						2197909643001732270
10	17352470721392640	224039677536202752	1316345575242829824	4647623631888061440	10962329653475101440	18152674910055544320	21563177468914464768	18418547421364438656	11137231418406342480	4585918819343788080	1188941916126167280	163929870253759428	7355827511386641					92385468660839479749
11	336637931995017216	4737224506940190720	30581415983691675648	119787447347097513984	317200312876360193280	598543199079740538624	825946708407527266560	840963921287664125952	628532930754061469136	337829353024992388560	125195583768085414584	29507376645676697040	3729404548273026987	154472377739119461				3863045988539844637752
12	6496765038089404416	98971552006535061504	696372002520208035840	2996978766401784213504	8804377380011667277824	18651378397129979364864	29328616754907286392576	34689761753116145195520	30905424107321656628736	20532075737966007991776	9932182978934776223664	3346136511619737444336	722930727819079077480	84341918245559225706	3243919932521508681			160799289272971033046427
13	124812851404832980992	2046480986998162391040	15588019441029272186880	73119060264621843763200	236012077854140506813440	554675774940735038502912	979197365849323916655360	1319787753970827887665920	1365909369028948217073600	1081689843756257982005760	646760385745929251740944	284421603487677682768560	87836083430018107913820	17517167635616146877400	1897693160525082578385	68122318582951682301		6666651614722636883600514
14	2388310778233020284928	41937118072023881613312	343808805815691281694720	1745858177395278484930560	6142001062228234876108800	15860011631798242057863168	31061843396681162156163072	47001473560767547999457280	55431085666774771281968640	50993949777080733437414400	36344778750210268195393536	19755589964602929871359744	7963804897654975117519680	2270224925575852635311040	420412023254787525057600	42508326795761849755824	1430568690241985328321	275423106964176735657224625

Technical details

Node id: n-9848ee772d97c947 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-d56b748473677c64

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:            1080,              1638,               441;
  1:           26244,             68040,             51597,               9261;
  2:          594864,           2204496,           2857680,            1444716,             194481;
  3:        12912048,          62460720,         115736040,          100018800,           37923795,            4084101;
  4:       272097792,        1626918048,        3935025360,         4860913680,         3150592200,          955679634,           85766121;
  5:      5612016960,       39998375424,      119578476528,       192816242640,       178638577740,        92627410680,        23414151033,         1801088541;
  6:    113872925952,      942818849280,     3359863535616,      6696394685568,      8098282190880,      6002256212064,      2593567499040,       561939624792,        37822859361;
  7:   2281131839232,    21521983004928,    89096381256960,    211671402743808,    316404648893088,    306115066815264,    189071070680016,     70026322474080,     13275823635711,      794280046581;
  8:  45225918203904,   479037686238720,  2259808215517440,   6236746687987200,  11112748644049920,  13288995253509696,  10714027338534240,   5672132120400480,   1838190964944600,   309769218166590,   16679880978201;
  9: 889046339429376, 10447187105101824, 55328852760572160, 174005232594842880, 360172121231260800, 513408987355106304, 511626317260123296, 353562902171629920, 163782814976563860, 47180234766911400, 7155668939648229, 350277500542221;

%F Formula

The row polynomials satisfy (7*n+13+7*x)*P(n,x) = (126*n+360+(147*n+546)*x+147*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1080 + 1638*x + 441*x^2.

%t Mathematica

P[0, x_] := 1080 + 1638*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((126*n + 360 + (147*n + 546)*x + 147*x^2)*P[n - 1, x])/(7*n + 13 + 7*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1080 + 1638*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((126*n + 360 + (147*n + 546)*x + 147*x^2)*v[i - 1])/(7*n + 13 + 7*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_A013610 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.