A166344 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

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

Rows ?

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

Technical details

Node id: n-943c738f9048f5dc · revision 0 · JSON
Definition source: derived:mobius_x_over_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: 1,    8,      8;
   2: 1,   18,     48,       32;
   3: 1,   38,    198,      320,       160;
   4: 1,   78,    708,     2072,      2400,        960;
   5: 1,  158,   2358,    11120,     22360,      20160,       6720;
   6: 1,  318,   7548,    53912,    167400,     255120,     188160,      53760;
   7: 1,  638,  23598,   245840,   1106560,    2535120,    3102960,    1935360,      483840;
   8: 1, 1278,  72708,  1077752,   6762000,   21850080,   39466560,   40306560,    21772800,    4838400;
   9: 1, 2558, 221958,  4601840,  39198760,  171672480,  429216480,  638184960,   558714240,  266112000,   53222400;
  10: 1, 5118, 673548, 19295192, 219003000, 1265227440, 4206222720, 8539211520, 10772092800, 8248262400, 3512678400, 638668800;

%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 + 8*x + 8*x^2.

%t Mathematica

P[1, x_] := 1 + 8*x + 8*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 + 8*x + 8*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
  - 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+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