A272867 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,x + 2\,x^{2} + 6\,n + 4\,n x\right)\,P_{n}(x) = \left(18 + 72\,x + 96\,x^{2} + 48\,x^{3} + 8\,x^{4} + 36\,n + 96\,n x + 72\,n x^{2} + 16\,n x^{3}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	18	20	4													42
2	180	528	480	160	16											1364
3	1512	7344	13392	11520	4896	960	64									39688
4	11664	79488	221184	324864	273024	133632	36864	5120	256							1086096
5	85536	751680	2825280	5944320	7706880	6400512	3432960	1167360	238080	25600	1024					28579232
6	606528	6531840	30979584	85224960	150854400	180320256	148746240	85045248	33315840	8663040	1400832	122880	4096			731815744
7	4199040	53561088	306996480	1045094400	2354075136	3701376000	4178635776	3432775680	2057287680	892588032	274821120	57802752	7741440	573440	16384	18367544448

All 15 rows

Technical details

Node id: n-e5977c2b696b104a · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-a6f3cf41f153a3d4

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:        18,         20,           4;
  2:       180,        528,         480,          160,           16;
  3:      1512,       7344,       13392,        11520,         4896,          960,            64;
  4:     11664,      79488,      221184,       324864,       273024,       133632,         36864,          5120,           256;
  5:     85536,     751680,     2825280,      5944320,      7706880,      6400512,       3432960,       1167360,        238080,         25600,          1024;
  6:    606528,    6531840,    30979584,     85224960,    150854400,    180320256,     148746240,      85045248,      33315840,       8663040,       1400832,       122880,         4096;
  7:   4199040,   53561088,   306996480,   1045094400,   2354075136,   3701376000,    4178635776,    3432775680,    2057287680,     892588032,     274821120,     57802752,      7741440,      573440,       16384;
  8:  28553472,  421023744,  2830712832,  11496038400,  31499145216,  61618765824,   88846958592,   96143376384,   78773575680,   48945364992,   22939435008,   8001552384,   2027225088,   357826560,    40894464,    2621440,     65536;
  9: 191476224, 3204707328, 24730665984, 116740030464, 377147695104, 884149862400, 1556354580480, 2100317257728, 2199822188544, 1799210631168, 1150320476160, 572671918080, 219906441216, 64061964288, 13780647936, 2095054848, 208797696, 11796480, 262144;

%F Formula

The row polynomials satisfy (6*n-3+(4*n+2)*x+2*x^2)*P(n,x) = (36*n+18+(96*n+72)*x+(72*n+96)*x^2+(16*n+48)*x^3+8*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((36*n + 18 + (96*n + 72)*x + (72*n + 96)*x^2 + (16*n + 48)*x^3 + 8*x^4)*P[n - 1, x])/(6*n - 3 + (4*n + 2)*x + 2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((36*n + 18 + (96*n + 72)*x + (72*n + 96)*x^2 + (16*n + 48)*x^3 + 8*x^4)*v[i - 1])/(6*n - 3 + (4*n + 2)*x + 2*x^2)); 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013609 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.