A128813 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 4\,n\right)\,P_{n}(x) = \left(-4 + 12\,n x - n^{2} - 6\,n^{2} x - 9\,n^{2} x^{2} - 3\,n^{3} - 6\,n^{3} x + 9\,n^{3} x^{2} + n^{4} + 6\,n^{4} x + 9\,n^{4} x^{2} - n^{5} - 6\,n^{5} x - 9\,n^{5} x^{2}\right)\,P_{n-2}(x) + \left(4\,n - 12\,n x + 4\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 5 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	3														8
2	38	84	27													149
3	410	1812	2187	486												4895
4	6190	42852	100287	85536	14580											249445
5	126760	1215012	4371687	7106211	4738500	656100										18214270
6	3409648	42081096	207313272	512028216	643792293	354294000	41334300									1804252825
7	117093680	1774958760	11180204712	37523468808	71117855253	73281287412	34431471900	3472081200								232898421725
8	5011583984	90286591176	695227393656	2968363066680	7603895780469	11744281574064	10282049078796	4222050739200	374984769600							37986150577625
9	261942062624	5474901670416	49940917502076	259761709396980	843350145857289	1753499683652139	2291575805392716	1747284258264660	637849093089600	50622943896000						7639621400784500
10	16434650493184	391269479222976	4131002270870856	25390006871642280	100152685791743094	263166432049616364	461573988255254871	523699599572496000	354091480500910500	116432770960800000	8352785742840000					1857398456145890125
11	1219789726118656	32604508560164928	391416491150283672	2779325732538009720	12929192443151864730	41190080864538474324	91095619994821948887	138383585765704072458	138890207133358690500	84426281672969079000	25267176872091000000	1653851577082320000				537040562845692026875
12	105759747692338432	3137221408747595712	42228068087198899368	340463108014975308168	1827032750722538616510	6853480761075653682108	18345773832096674475243	35078054661219614694912	47048133277989351507672	42370680346251232104000	23407362313029355806000	6430174931696060160000	387001269037262880000			182133628300376358068125
13	10595927905848040448	345503160726642684672	5155671240903511442304	46555546877020421364384	283343879088624528259884	1224594071648851246240392	3854762498175050071943883	8908783741896580907492571	15017624992289346473903256	18045715485267286497174456	14747045389925837716782000	7469407604160002898558000	1896693219551625374880000	105651346447172766240000		71605689545656934905006250
14	1215100892100739094528	43206837083627543250432	708128502600980238424704	7082671938331534277102784	48234760884100300030735824	236203895403389370322636272	855785879154892405316180448	2323269301444205751361952496	4728332482982518160013247881	7132325993728223140353371616	7767134399005378906012505640	5812146693686925917066448000	2718728977040462718117450000	641937581013021727674240000	33280174130859421365600000	32305215360852886060432240625

Technical details

Node id: n-27cca71ec864560f · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-f5f6de77b771b81b

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:            5,             3;
  2:           38,            84,             27;
  3:          410,          1812,           2187,             486;
  4:         6190,         42852,         100287,           85536,           14580;
  5:       126760,       1215012,        4371687,         7106211,         4738500,           656100;
  6:      3409648,      42081096,      207313272,       512028216,       643792293,        354294000,         41334300;
  7:    117093680,    1774958760,    11180204712,     37523468808,     71117855253,      73281287412,      34431471900,       3472081200;
  8:   5011583984,   90286591176,   695227393656,   2968363066680,   7603895780469,   11744281574064,   10282049078796,    4222050739200,    374984769600;
  9: 261942062624, 5474901670416, 49940917502076, 259761709396980, 843350145857289, 1753499683652139, 2291575805392716, 1747284258264660, 637849093089600, 50622943896000;

%F Formula

The row polynomials satisfy (4*n-4)*P(n,x) = -(n^5-n^4+3*n^3+n^2+4+(6*n^5-6*n^4+6*n^3+6*n^2-12*n)*x+(9*n^5-9*n^4-9*n^3+9*n^2)*x^2)*P(n-2,x) + (4*n^3+4*n+(12*n^3-12*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (6*n^5 - 6*n^4 + 6*n^3 + 6*n^2 - 12*n)*x + (9*n^5 - 9*n^4 - 9*n^3 + 9*n^2)*x^2)*P[n - 2, x] + (4*n^3 + 4*n + (12*n^3 - 12*n)*x)*P[n - 1, x])/(4*n - 4)]];
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] = 5 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (6*n^5 - 6*n^4 + 6*n^3 + 6*n^2 - 12*n)*x + (9*n^5 - 9*n^4 - 9*n^3 + 9*n^2)*x^2)*v[i - 2] + (4*n^3 + 4*n + (12*n^3 - 12*n)*x)*v[i - 1])/(4*n - 4)); 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 of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
    
    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 coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).?
    
    ↑ 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.