A038333 → 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

\(\left(40 + 21\,x + 21\,n\right)\,P_{n}(x) = \left(1159 + 1680\,x + 441\,x^{2} + 399\,n + 441\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1159 + 1680\,x + 441\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1159	1680	441															3280
1	29602	73017	52920	9261														164800
2	706477	2486568	3066714	1481760	194481													7936000
3	16159804	74180085	130544820	107334990	38896200	4084101												371200000
4	359034355	2036135304	4673345355	5482882440	3381052185	980184240	85766121											16998400000
5	7809616246	52778050185	149655944844	228993922395	201495929670	99402934239	24014513880	1801088541										765952000000
6	167154015193	1312015529328	4433356215540	8380732911264	9617744740590	6770263236912	2783282158692	576348333120	37822859361									34078720000000
7	3532581112528	31592108871477	123985467521496	279301441579020	395989630057224	363550751194302	213263291962728	75148618284684	13616229369960	794280046581								1500774400000000
8	73895482791391	741842033630880	3317171431505085	8678982726504720	14663325682898550	16631564462403408	12724276291800570	6397898758881840	1972651229972955	317712018632400	16679880978201							65536000000000000
9	1532766564450250	17069856524811321	85682754884366640	255422200225891545	501211252455647580	677445646549913010	640315231802531208	419901117629418810	184739326662713130	50631381569305845	7339147630408440	350277500542221						2841640960000000000
10	31568860161417349	386257174241463000	2150801922126226446	7197351410286797760	16091598614231167335	25261047123764638032	28452717155096346420	23051348344891123488	13226885205326692515	5172701146555967640	1275910815546507294	168133200260266080	7355827511386641					122473676800000000000
11	646287956367319012	8618298824066936277	52724104283959699500	195722974913486606586	491219233752073947120	878601284337021736491	1149377644131291030456	1109655969048757510380	786627262269409589028	401215517894909672955	141214741300977916572	31665786604017862842	3825030305921053320	154472377739119461				5251268608000000000000
12	13162583823686459467	190008659171991789528	1266889927137839632719	5166962219828050551000	14385638656141265584071	28883690944621948090656	43051462932514065088059	48273861053514223279152	40779856862541838506465	25696490567467379908248	11795736226110344384877	3774284903862500679288	775811771798437639629	86504531533906898160	3243919932521508681			224143605760000000000000
13	266868233051483296414	4146213904461234732105	29926363819588706850660	133023442349473161435495	406898274811458980891250	906295235336899731796473	1516393774592652274759440	1937315831963132928962655	1900783278982122541616610	1427294990188964347726275	809439452875222467109812	337786991929523498294205	99074978726390642831310	18798516008962142806395	1946351959512905208600	68122318582951682301		9529458688000000000000000
14	5389300095605553396145	89667726305298387595104	696563935949487434993640	3351752747793935167273920	11173969157355745560581580	27343564067330043515892000	50752533178866384980602488	72786901180447309188453120	81367264942451583016431510	70962575748665908220353440	47957111670349202083602840	24724696015097704449899712	9458035774026657952237740	2560707142466711999332320	451164384215091427353480	43598283893089076672640	1430568690241985328321	403726925824000000000000000

Technical details

Node id: n-cdf36e798942c344 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-9d4c93c35b848be1

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:             1159,              1680,               441;
  1:            29602,             73017,             52920,               9261;
  2:           706477,           2486568,           3066714,            1481760,             194481;
  3:         16159804,          74180085,         130544820,          107334990,           38896200,            4084101;
  4:        359034355,        2036135304,        4673345355,         5482882440,         3381052185,          980184240,           85766121;
  5:       7809616246,       52778050185,      149655944844,       228993922395,       201495929670,        99402934239,        24014513880,         1801088541;
  6:     167154015193,     1312015529328,     4433356215540,      8380732911264,      9617744740590,      6770263236912,      2783282158692,       576348333120,        37822859361;
  7:    3532581112528,    31592108871477,   123985467521496,    279301441579020,    395989630057224,    363550751194302,    213263291962728,     75148618284684,     13616229369960,      794280046581;
  8:   73895482791391,   741842033630880,  3317171431505085,   8678982726504720,  14663325682898550,  16631564462403408,  12724276291800570,   6397898758881840,   1972651229972955,   317712018632400,   16679880978201;
  9: 1532766564450250, 17069856524811321, 85682754884366640, 255422200225891545, 501211252455647580, 677445646549913010, 640315231802531208, 419901117629418810, 184739326662713130, 50631381569305845, 7339147630408440, 350277500542221;

%F Formula

The row polynomials satisfy (21*n+40+21*x)*P(n,x) = (399*n+1159+(441*n+1680)*x+441*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1159 + 1680*x + 441*x^2.

%t Mathematica

P[0, x_] := 1159 + 1680*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((399*n + 1159 + (441*n + 1680)*x + 441*x^2)*P[n - 1, x])/(21*n + 40 + 21*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1159 + 1680*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((399*n + 1159 + (441*n + 1680)*x + 441*x^2)*v[i - 1])/(21*n + 40 + 21*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)*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.