A038331 → 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

\(\left(32 + 15\,x + 15\,n\right)\,P_{n}(x) = \left(799 + 960\,x + 225\,x^{2} + 255\,n + 225\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 799 + 960\,x + 225\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	799	960	225								1984
1	17918	35955	21600	3375							78848
2	378301	1075080	1078650	432000	50625						3014656
3	7683932	28372575	40315500	26966250	8100000	759375					112197632
4	151924699	691553880	1276765875	1209465000	606740625	145800000	11390625				4093640704
5	2944783418	15952093395	36306578700	44686805625	31748456250	12741553125	2551500000	170859375			147102629888
6	56216398201	353374010160	957125603700	1452263148000	1340604168750	761962950000	254831062500	43740000000	2562890625		5222680231936
7	1060315131032	7589213757135	23852745685800	43070652166500	49013881245000	36196312556250	17144166375000	4914599062500	738112500000	38443359375	183618441838592

All 15 rows

Technical details

Node id: n-3e7c547c780b8c12 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-235bca8d2225d93d

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:             799,              960,               225;
  1:           17918,            35955,             21600,              3375;
  2:          378301,          1075080,           1078650,            432000,             50625;
  3:         7683932,         28372575,          40315500,          26966250,           8100000,            759375;
  4:       151924699,        691553880,        1276765875,        1209465000,         606740625,         145800000,          11390625;
  5:      2944783418,      15952093395,       36306578700,       44686805625,       31748456250,       12741553125,        2551500000,         170859375;
  6:     56216398201,     353374010160,      957125603700,     1452263148000,     1340604168750,      761962950000,      254831062500,       43740000000,       2562890625;
  7:   1060315131032,    7589213757135,    23852745685800,    43070652166500,    49013881245000,    36196312556250,    17144166375000,     4914599062500,     738112500000,      38443359375;
  8:  19804175374999,  159047269654800,   569191031785125,  1192637284290000,  1615149456243750,  1470416437350000,   904907813906250,   367374993750000,   92148732421875,   12301875000000,    576650390625;
  9: 366910889881718, 3267688936874835, 13121399746521000, 31305506748181875, 49196287976962500, 53299932056043750, 40436452027125000, 21329969899218750, 7577109246093750, 1689393427734375, 202980937500000, 8649755859375;

%F Formula

The row polynomials satisfy (15*n+32+15*x)*P(n,x) = (255*n+799+(225*n+960)*x+225*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 799 + 960*x + 225*x^2.

%t Mathematica

P[0, x_] := 799 + 960*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((255*n + 799 + (225*n + 960)*x + 225*x^2)*P[n - 1, x])/(15*n + 32 + 15*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 799 + 960*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((255*n + 799 + (225*n + 960)*x + 225*x^2)*v[i - 1])/(15*n + 32 + 15*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)*5^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    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
    
    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.