A166343 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

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

\(P_{1}(x) = 1 + 14\,x + 14\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	1	14	14															29
2	1	30	84	56														171
3	1	62	342	560	280													1245
4	1	126	1212	3608	4200	1680												10827
5	1	254	4014	19280	39040	35280	11760											109629
6	1	510	12804	93176	291600	445920	329280	94080										1267371
7	1	1022	39942	423920	1923880	4425120	5426400	3386880	846720									16473885
8	1	2046	122892	1855448	11739000	38094000	68960640	70506240	38102400	8467200								237849867
9	1	4094	374814	7913360	67972240	298998000	749382480	1115735040	977477760	465696000	93139200							3776692989
10	1	8190	1136724	33152696	379428000	2201821440	7338663360	14920940160	18838915200	14431737600	6147187200	1117670400						65410660971
11	1	16382	3434742	137157680	2062903480	15487496640	66783393600	178076828160	303838698240	332706528000	226368172800	87178291200	14529715200					1227172636125
12	1	32766	10353372	562369688	11000305800	105302400720	575896231680	1958881774080	4337239737600	6365452262400	6149821708800	3762557568000	1322204083200	203416012800				24792344840907
13	1	65534	31158414	2290892240	57813377440	697816239120	4768390426800	20278224046080	56665093605120	107026920000000	137668013683200	118948551321600	66101901465600	21358681344000	3051240192000			536624967817149
14	1	131070	93671844	9288202616	300521348400	4533777699360	38263446661440	200372915783040	692489858860800	1636920136051200	2691644270515200	3079398780057600	2405655886233600	1224448159334400	366148823040000	48819843072000		12389005800662571
15	1	262142	281408742	37527497840	1549047755080	29005794286560	299580570525600	1909090899555840	8035764971794560	23294099949120000	47614208472230400	69252516606873600	71305710661785600	50821456637952000	23858954735616000	6639498657792000	829937332224000	303891412146679965

Technical details

Node id: n-5e5f6e74cb22a352 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-addc5ba4413954b4

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 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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1,   14,      14;
   2: 1,   30,      84,       56;
   3: 1,   62,     342,      560,       280;
   4: 1,  126,    1212,     3608,      4200,       1680;
   5: 1,  254,    4014,    19280,     39040,      35280,      11760;
   6: 1,  510,   12804,    93176,    291600,     445920,     329280,       94080;
   7: 1, 1022,   39942,   423920,   1923880,    4425120,    5426400,     3386880,      846720;
   8: 1, 2046,  122892,  1855448,  11739000,   38094000,   68960640,    70506240,    38102400,     8467200;
   9: 1, 4094,  374814,  7913360,  67972240,  298998000,  749382480,  1115735040,   977477760,   465696000,   93139200;
  10: 1, 8190, 1136724, 33152696, 379428000, 2201821440, 7338663360, 14920940160, 18838915200, 14431737600, 6147187200, 1117670400;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + 14*x + 14*x^2.

%t Mathematica

P[1, x_] := 1 + 14*x + 14*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + 14*x + 14*x^2; for(n = 2, N, my(i = n); v[i] = ((1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with …?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Parent-row provers of polytool polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with …?
    
    ↑ Liu–Wang theorem (above)
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A166343 → x ↦ x+1

References

[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