A144699 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 8\,x + 13\,x^{2} + 6\,x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	6														7
2	1	18	42													61
3	1	42	252	336												631
4	1	90	1050	3360	3024											7525
5	1	186	3780	21840	45360	30240										101407
6	1	378	12642	117600	423360	635040	332640									1521661
7	1	762	40572	571536	3175200	8043840	9313920	3991680								25137511
8	1	1530	127050	2610720	21019824	80015040	153679680	143700480	51891840							453046165
9	1	3066	391860	11459280	128595600	690288480	1955923200	2993760000	2335132800	726485760						8842040047
10	1	6138	1197042	48972000	746111520	5427686880	21284635680	47421158400	59935075200	39956716800	10897286400					185718846061
11	1	12282	3634092	205464336	4171305600	40027236480	208697005440	634784895360	1155890736000	1238658220800	719220902400	174356582400				4176015995191
12	1	24570	10988250	850930080	22705707024	281876474880	1901178639360	7582643228160	18655220263680	28569052512000	26491303238400	13599813427200	2964061900800			100068717334405
13	1	49146	33136740	3491626320	121186905840	1918315919520	16408891699200	83475289497600	266471344339200	546863608811520	719940123302400	587058612940800	269729632972800	53353114214400		2545343645415487
14	1	98298	99754242	14231599200	637359166080	12721764575520	135963717009120	864709016371200	3483420862521600	9199234908864000	16122295488499200	18563745328128000	13486481648640000	5602076992512000	1013709170073600	68485010587812061

Technical details

Node id: n-8d95b187b41dd93e · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-bc84dda52d053345

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 1,    6;
  2: 1,   18,     42;
  3: 1,   42,    252,      336;
  4: 1,   90,   1050,     3360,      3024;
  5: 1,  186,   3780,    21840,     45360,     30240;
  6: 1,  378,  12642,   117600,    423360,    635040,     332640;
  7: 1,  762,  40572,   571536,   3175200,   8043840,    9313920,    3991680;
  8: 1, 1530, 127050,  2610720,  21019824,  80015040,  153679680,  143700480,   51891840;
  9: 1, 3066, 391860, 11459280, 128595600, 690288480, 1955923200, 2993760000, 2335132800, 726485760;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+8*x+13*x^2+6*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 8*x + 13*x^2 + 6*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, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 + 8*x + 13*x^2 + 6*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 of 5-Eulerian numbers.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

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