A166341 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

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

Technical details

Node id: n-4654cca59858ef1f · revision 0 · JSON
Definition source: derived:shift_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:        12,         12,           1;
   2:        48,         72,          26,           1;
   3:       240,        480,         294,          54,           1;
   4:      1440,       3600,        3096,        1044,         110,          1;
   5:     10080,      30240,       33480,       16560,        3462,        222,          1;
   6:     80640,     282240,      382320,      250200,       80088,      11052,        446,         1;
   7:    725760,    2903040,     4651920,     3795120,     1651440,     364560,      34494,       894,        1;
   8:   7257600,   32659200,    60439680,    59129280,    32679360,   10080000,    1596216,    106164,     1790,      1;
   9:  79833600,  399168000,   837889920,   956551680,   642660480,  256556160,   58381080,   6809520,   323862,   3582,    1;
  10: 958003200, 5269017600, 12370579200, 16149974400, 12793697280, 6294516480, 1889623440, 325953000, 28533528, 982332, 7166, 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) = 12 + 12*x + x^2.

%t Mathematica

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

A166341 → 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