A166344 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

\(P_{1}(x) = 8 + 8\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
1	8	8	1															17
2	32	48	18	1														99
3	160	320	198	38	1													717
4	960	2400	2072	708	78	1												6219
5	6720	20160	22360	11120	2358	158	1											62877
6	53760	188160	255120	167400	53912	7548	318	1										726219
7	483840	1935360	3102960	2535120	1106560	245840	23598	638	1									9433917
8	4838400	21772800	40306560	39466560	21850080	6762000	1077752	72708	1278	1								136148139
9	53222400	266112000	558714240	638184960	429216480	171672480	39198760	4601840	221958	2558	1							2161147677
10	638668800	3512678400	8248262400	10772092800	8539211520	4206222720	1265227440	219003000	19295192	673548	5118	1						37421340939
11	8302694400	49816166400	129370348800	190203552000	173801738880	101963473920	38300151120	8905382640	1191490960	79874960	2035998	10238	1					701936920317
12	116237721600	755545190400	2150238182400	3515312908800	3640052908800	2481886915200	1122109000320	330438736320	60581241600	6356829600	327643832	6138708	20478	1				14179093438059
13	1743565824000	12204960768000	37775183846400	67986613094400	78709023993600	61219398240000	32435963239680	11620381893120	2737139845440	401628427200	33422367160	1335130160	18477558	40958	1			306868635187677
14	27897053184000	209227898880000	699722024601600	1374903360230400	1760347645056000	1539212644569600	936553614796800	396507106195200	114860173908480	21971377908480	2610304766160	173787486600	5414430872	55555548	81918	1		7083992461651659
15	474249904128000	3793999233024000	13634248852224000	29044755387648000	40758263068723200	39594723475507200	27233428853030400	13330607209920000	4602305520933120	1094652414535680	172071778722480	16704553516560	896009587360	21879945680	166912398	163838	1	173750928308521917

Technical details

Node id: n-25a1bf976309f071 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-ea33b747599ac881

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:         8,          8,          1;
   2:        32,         48,         18,           1;
   3:       160,        320,        198,          38,          1;
   4:       960,       2400,       2072,         708,         78,          1;
   5:      6720,      20160,      22360,       11120,       2358,        158,          1;
   6:     53760,     188160,     255120,      167400,      53912,       7548,        318,         1;
   7:    483840,    1935360,    3102960,     2535120,    1106560,     245840,      23598,       638,        1;
   8:   4838400,   21772800,   40306560,    39466560,   21850080,    6762000,    1077752,     72708,     1278,      1;
   9:  53222400,  266112000,  558714240,   638184960,  429216480,  171672480,   39198760,   4601840,   221958,   2558,    1;
  10: 638668800, 3512678400, 8248262400, 10772092800, 8539211520, 4206222720, 1265227440, 219003000, 19295192, 673548, 5118, 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) = 8 + 8*x + x^2.

%t Mathematica

P[1, x_] := 8 + 8*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] = 8 + 8*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
  - 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 polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
  - ↑ Parent-row provers of polytool (above)
  - ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A166344 → 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