A166341 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

\(P_{1}(x) = 1 + 12\,x + 12\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	1	12	12															25
2	1	26	72	48														147
3	1	54	294	480	240													1069
4	1	110	1044	3096	3600	1440												9291
5	1	222	3462	16560	33480	30240	10080											94045
6	1	446	11052	80088	250200	382320	282240	80640										1086987
7	1	894	34494	364560	1651440	3795120	4651920	2903040	725760									14127229
8	1	1790	106164	1596216	10080000	32679360	59129280	60439680	32659200	7257600								203949291
9	1	3582	323862	6809520	58381080	256556160	642660480	956551680	837889920	399168000	79833600							3238177885
10	1	7166	982332	28533528	325953000	1889623440	6294516480	12793697280	16149974400	12370579200	5269017600	958003200						56080887627
11	1	14334	2968494	118063440	1772432640	13293458640	57288979440	152705710080	260493045120	285205536000	194035564800	74724249600	12454041600					1052094064189
12	1	28670	8948484	484127736	9452480400	90395347680	494077066560	1679957516160	3718788796800	5456985811200	5271652108800	3225117772800	1133317785600	174356582400				21254594373291
13	1	57342	26931462	1972304880	49683040680	599086968480	4091306899680	17392276661760	48588716816640	91757746080000	118015017120000	101961238579200	56659662259200	18307441152000	2615348736000			460039523607325
14	1	114686	80966412	7996945368	258276727800	3892620054960	32832757077120	171868668491520	593828941305600	1403464628966400	2307500395200000	2639715068390400	2062071710899200	1049539447756800	313841848320000	41845579776000		10620668020992267
15	1	229374	243243294	32311647120	1331368365840	24905380696560	257077639924560	1637611404549120	6891278488174080	19972935702720000	40820615265830400	59366585563084800	61123228130764800	43562556221184000	20450719441152000	5690998849536000	711374856192000	260511250867293949

Technical details

Node id: n-83613b7858f852b2 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-b6466966904f5921

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,   12,     12;
   2: 1,   26,     72,       48;
   3: 1,   54,    294,      480,       240;
   4: 1,  110,   1044,     3096,      3600,       1440;
   5: 1,  222,   3462,    16560,     33480,      30240,      10080;
   6: 1,  446,  11052,    80088,    250200,     382320,     282240,       80640;
   7: 1,  894,  34494,   364560,   1651440,    3795120,    4651920,     2903040,      725760;
   8: 1, 1790, 106164,  1596216,  10080000,   32679360,   59129280,    60439680,    32659200,     7257600;
   9: 1, 3582, 323862,  6809520,  58381080,  256556160,  642660480,   956551680,   837889920,   399168000,   79833600;
  10: 1, 7166, 982332, 28533528, 325953000, 1889623440, 6294516480, 12793697280, 16149974400, 12370579200, 5269017600, 958003200;

%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 + 12*x + 12*x^2.

%t Mathematica

P[1, x_] := 1 + 12*x + 12*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 + 12*x + 12*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 ?

A166341 → 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