A008955 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-4\,n^{2} - 8\,n^{2} x - 4\,n^{2} x^{2} + 8\,n^{3} + 24\,n^{3} x + 16\,n^{3} x^{2} - 8\,n^{4} - 28\,n^{4} x - 24\,n^{4} x^{2} + 4\,n^{5} + 16\,n^{5} x + 16\,n^{5} x^{2} - n^{6} - 4\,n^{6} x - 4\,n^{6} x^{2}\right)\,P_{n-2}(x) + \left(1 - 2\,n + 4\,n^{2} + 4\,n^{2} x - 4\,n^{3} - 8\,n^{3} x + 2\,n^{4} + 4\,n^{4} x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	26	32													58
3	440	1256	864												2560
4	10680	49512	74976	36864											172032
5	362680	2355312	5662776	5956864	2304000										16641632
6	16601560	139372464	463896408	764030464	621365760	199065600									2204332256
7	990347200	10222528960	43682572608	98817320512	124662019840	83025893376	22759833600								384160516096
8	74839128000	917993265600	4802092779840	13878284891712	23913802571520	24544769316864	13877922594816	3329438515200							85339143063552
9	6997846596000	99523325251200	616843971487680	2175271401565824	4771284501436320	6661514303886336	5776951053662208	2842412198658048	606790169395200						23557588771938816
10	793949326196000	12850994488771200	92158238695615680	384194137288196224	1025717585036524320	1817921225037236736	2137825393993361408	1607529587187253248	700828551806976000	134842259865600000					7914661922725730816
11	107514474868112000	1952092801508950400	15908216120289058560	76608076434945265408	241356271904135324800	519654776149149544320	774074457269369930240	787380032889865994240	523151013432180768768	204895766300590080000	35895009576222720000				3180983227353125748736
12	17136259016573840000	344981793742830320000	3149995427693780064000	17216926073258061971200	62575566942679917642880	158762670111695852177280	286844801090172886776320	368951183275977471918080	330968027603982793605120	197118210411883407605760	70115755152579231744000	11277559372311429120000			1507342813514994236784640
13	3176364777525154800000	70231016031485070240000	710404992135557583120000	4346475371867867417280000	17911997646207117182121600	52370303870939269466089728	111369029099374788460548480	173525750035177239632240640	196558252133367508609720320	157806977247960384002457600	85208229007076720344301568	27770776265916703309824000	4129466323494701629440000		831781069374326867862183936
14	677636457702972055600000	16325780143782280200480000	181245620877218065175760000	1227560657480821311314880000	5658436468518057119345155200	18742753555350864007289677056	45890384369553059229219752832	84077897793449821437877125120	115246831435786561873933025280	116700013224826043842940108800	84832389703714631489687126016	41908980128866398961089380352	12606827853321082893238272000	1743270091026070964797440000	528833594319372116448163782656

Technical details

Node id: n-f81aa065f11fbdb2 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-a8d9942311e374e2

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
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:             0;
  1:             2;
  2:            26,             32;
  3:           440,           1256,             864;
  4:         10680,          49512,           74976,            36864;
  5:        362680,        2355312,         5662776,          5956864,          2304000;
  6:      16601560,      139372464,       463896408,        764030464,        621365760,        199065600;
  7:     990347200,    10222528960,     43682572608,      98817320512,     124662019840,      83025893376,      22759833600;
  8:   74839128000,   917993265600,   4802092779840,   13878284891712,   23913802571520,   24544769316864,   13877922594816,    3329438515200;
  9: 6997846596000, 99523325251200, 616843971487680, 2175271401565824, 4771284501436320, 6661514303886336, 5776951053662208, 2842412198658048, 606790169395200;

%F Formula

The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^6-4*n^5+8*n^4-8*n^3+4*n^2+(4*n^6-16*n^5+28*n^4-24*n^3+8*n^2)*x+(4*n^6-16*n^5+24*n^4-16*n^3+4*n^2)*x^2)*P(n-2,x) + (2*n^4-4*n^3+4*n^2-2*n+1+(4*n^4-8*n^3+4*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (4*n^6 - 16*n^5 + 28*n^4 - 24*n^3 + 8*n^2)*x + (4*n^6 - 16*n^5 + 24*n^4 - 16*n^3 + 4*n^2)*x^2)*P[n - 2, x] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (4*n^4 - 8*n^3 + 4*n^2)*x)*P[n - 1, x])/(n^2 - 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (4*n^6 - 16*n^5 + 28*n^4 - 24*n^3 + 8*n^2)*x + (4*n^6 - 16*n^5 + 24*n^4 - 16*n^3 + 4*n^2)*x^2)*v[i - 2] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (4*n^4 - 8*n^3 + 4*n^2)*x)*v[i - 1])/(n^2 - 2*n + 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 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_A013609 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)

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.