A038305 → 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

\(\left(22 + 9\,x + 9\,n\right)\,P_{n}(x) = \left(403 + 396\,x + 81\,x^{2} + 117\,n + 81\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 403 + 396\,x + 81\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	403	396	81								880
1	6760	10881	5346	729							23716
2	107653	243360	195858	64152	6561						617584
3	1656538	4844385	5475600	2937870	721710	59049					15695152
4	24876631	89453052	130798395	98560800	39661245	7794468	531441				391676032
5	366837484	1567227753	2817771138	2746766295	1552332600	499731687	81841914	4782969			9637291840
6	5333623945	26412298848	56420199108	67626507312	49441793310	22353589440	5996780244	841802544	43046721		234469641472
7	76678687774	432023539545	1069698103344	1523345375916	1369436773068	800957051622	301773457440	69391314252	8523250758	387420489	5652214974208

All 15 rows

Technical details

Node id: n-599179c3856533fe · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-e470dede5399547f

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:            403,             396,              81;
  1:           6760,           10881,            5346,             729;
  2:         107653,          243360,          195858,           64152,            6561;
  3:        1656538,         4844385,         5475600,         2937870,          721710,           59049;
  4:       24876631,        89453052,       130798395,        98560800,        39661245,         7794468,          531441;
  5:      366837484,      1567227753,      2817771138,      2746766295,      1552332600,       499731687,        81841914,         4782969;
  6:     5333623945,     26412298848,     56420199108,     67626507312,     49441793310,     22353589440,      5996780244,       841802544,       43046721;
  7:    76678687774,    432023539545,   1069698103344,   1523345375916,   1369436773068,    800957051622,    301773457440,     69391314252,     8523250758,     387420489;
  8:  1092263435419,   6901081899660,  19441059279525,  32090943100320,  34275270958110,  24649861915224,  12014355774330,   3879944452800,   780652285335,   85232507580,   3486784401;
  9: 15440151087088, 108134080106481, 341603554033170, 641554956224325, 794250841732920, 678650364970578, 406722721601196, 169917317379810, 48014312603400, 8587175138685, 843801825042, 31381059609;

%F Formula

The row polynomials satisfy (9*n+22+9*x)*P(n,x) = (117*n+403+(81*n+396)*x+81*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 403 + 396*x + 81*x^2.

%t Mathematica

P[0, x_] := 403 + 396*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((117*n + 403 + (81*n + 396)*x + 81*x^2)*P[n - 1, x])/(9*n + 22 + 9*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 403 + 396*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((117*n + 403 + (81*n + 396)*x + 81*x^2)*v[i - 1])/(9*n + 22 + 9*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)*10^(i-j)*3^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool 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.