A357367 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2\,n - n^{2}\right)\,P_{n-2}(x) + \left(-3 - 6\,x + 6\,n + 12\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 2 + 6\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	6														8
2	18	90	108													216
3	264	1872	4320	3240												9696
4	5400	49680	168480	249480	136080											609120
5	141840	1604880	7166880	15785280	17146080	7348320										49193280
6	4551120	61130160	338385600	987940800	1604383200	1374135840	484989120									4855515840
7	172529280	2682892800	17714350080	64371283200	139030214400	178475996160	126097171200	37829151360								566373588480
8	7545363840	133323563520	1022363596800	4443578092800	11972764742400	20478180602880	21713932880640	13051057219200	3404623622400							76227369684480
9	373944211200	7400106604800	64623542860800	326848178764800	1055097063820800	2254365226713600	3188240876620800	2878041835468800	1504843641100800	347271609484800						11627106025651200
10	20711190931200	453769831468800	4445365008268800	25641943828300800	96443403837580800	247141628750016000	436990251182284800	526463759978956800	413600486896396800	191346656826124800	39588963481267200					1982136940811596800
11	1267784551756800	30464498885990400	330835263542784000	2143200482527104000	9202364754815232000	27498609337341542400	58354595200200499200	87943062385930752000	92242284911352576000	64134120839652864000	26603783459411558400	4988209398639667200				373472798316852326400
12	84991791159475200	2222552311499520000	26498290229726515200	190459066403878272000	919151626210481664000	3137673384991066521600	7768836327225093120000	14057829791059415961600	18451228209714203904000	17131728056883568128000	10681419058953740697600	4015508565904932096000	688372897012274073600			77071012818691039948800
13	6193091146059417600	175083768695688652800	2273445171267798528000	17952695845748471808000	96189293766429287424000	369265946670662398156800	1044969025176913290854400	2207086825492032067584000	3479325937645154268672000	4043392656443327835648000	3367061296919369918668800	1902662687341925539430400	653954252161660369920000	103255934551841111040000		17287571274046174105804800
14	487361761916020992000	14811677241680487168000	208060916642799187968000	1790469358095158028288000	10545252048888663794688000	44966078141806699637760000	143158409897003469416448000	345697299538616568711168000	636273660339734704455168000	888287476756868106766848000	925967096992383844349952000	698905002336561866926080000	361086003127788365306880000	114304319548888109921280000	16727461397398259988480000	4187931889439680212999168000

Technical details

Node id: n-97cba47fa866c4c1 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-bbe4c1ef283db3b0

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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:            1;
  1:            2,             6;
  2:           18,            90,            108;
  3:          264,          1872,           4320,            3240;
  4:         5400,         49680,         168480,          249480,           136080;
  5:       141840,       1604880,        7166880,        15785280,         17146080,          7348320;
  6:      4551120,      61130160,      338385600,       987940800,       1604383200,       1374135840,        484989120;
  7:    172529280,    2682892800,    17714350080,     64371283200,     139030214400,     178475996160,     126097171200,      37829151360;
  8:   7545363840,  133323563520,  1022363596800,   4443578092800,   11972764742400,   20478180602880,   21713932880640,   13051057219200,    3404623622400;
  9: 373944211200, 7400106604800, 64623542860800, 326848178764800, 1055097063820800, 2254365226713600, 3188240876620800, 2878041835468800, 1504843641100800, 347271609484800;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2-2*n)*P(n-2,x) + (6*n-3+(12*n-6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n)*P[n - 2, x] + (6*n - 3 + (12*n - 6)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n)*v[i - 2] + (6*n - 3 + (12*n - 6)*x)*v[i - 1]); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows. T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.?
  - 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 for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

A386789 → basis A013610

References

[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.