A166343 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

Rows ?

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

Technical details

Node id: n-5de7394bf8d53953 · revision 0 · JSON
Definition source: derived:shift_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:         14,         14,           1;
   2:         56,         84,          30,           1;
   3:        280,        560,         342,          62,           1;
   4:       1680,       4200,        3608,        1212,         126,          1;
   5:      11760,      35280,       39040,       19280,        4014,        254,          1;
   6:      94080,     329280,      445920,      291600,       93176,      12804,        510,         1;
   7:     846720,    3386880,     5426400,     4425120,     1923880,     423920,      39942,      1022,        1;
   8:    8467200,   38102400,    70506240,    68960640,    38094000,   11739000,    1855448,    122892,     2046,       1;
   9:   93139200,  465696000,   977477760,  1115735040,   749382480,  298998000,   67972240,   7913360,   374814,    4094,    1;
  10: 1117670400, 6147187200, 14431737600, 18838915200, 14920940160, 7338663360, 2201821440, 379428000, 33152696, 1136724, 8190, 1;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 14 + 14*x + x^2; for(n = 2, N, my(i = n); v[i] = (n + 2 + (n + 1)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 1], x)); 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]
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)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by 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/(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