A290315 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 - 7\,x - 21\,x^{2} - 17\,x^{3} - 4\,x^{4} + 6\,n x + 14\,n x^{2} + 10\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x) + \left(2\,x + 4\,x^{2} - 4\,x^{4} - 2\,x^{5}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	3														4
3	1	18	29													48
4	1	69	419	471												960
5	1	228	3550	12372	10729											26880
6	1	711	23914	183822	444805	314427										967680
7	1	2166	143043	2081604	10212855	18872550	11265701									42577920
8	1	6537	799701	20171565	174242595	618389019	923334983	477107439								2214051840
9	1	19656	4292684	177265944	2483900790	14804568504	40876316204	51180241896	23316504721							132843110400
10	1	59019	22465892	1460063676	31447182462	291301853274	1306296310676	2941223165868	3170078181769	1291502224563						9033331507200
11	1	177114	115693561	11501945880	366562794178	5014330550460	33900529500026	120907652022744	229378827408605	216997891291002	79955783163629					686533194547200
12	1	531405	589626007	87805503555	4025538835018	78412119330594	761008461709454	3999491444347398	11787826379379365	19299271973407617	16267453907403755	5471210121890631				57668788341964800
13	1	1594284	2984797002	655333347468	42299042528583	1141700380309656	15359903779724140	113442900955130424	483968042726514207	1212042119226700092	1743993256496880618	1325489240307974076	410048406225261049			5305528527460761600
14	1	4782927	15043556334	4811228281674	429901079757387	15742928373304365	285983062340095348	2869620103130602044	16919993143342212999	60460832479347171513	131398761670249113534	168557999479709650650	116638876204717104397	33404608947510526827		530552852746076160000
15	1	14348862	75605198783	34897240825020	4259150166871425	208090424132329842	5000695228094526055	66460816450479623688	524082874663157721419	2553710702955882736194	7825901357335774406757	15003820343158277533308	17357562432995903993779	11023905412083708283086	2939051077157786881781	57299708096576225280000

Technical details

Node id: n-d90f64c532a1088f · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-2c9690c435047e3f

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 -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;
   2: 1,     3;
   3: 1,    18,       29;
   4: 1,    69,      419,        471;
   5: 1,   228,     3550,      12372,       10729;
   6: 1,   711,    23914,     183822,      444805,       314427;
   7: 1,  2166,   143043,    2081604,    10212855,     18872550,      11265701;
   8: 1,  6537,   799701,   20171565,   174242595,    618389019,     923334983,     477107439;
   9: 1, 19656,  4292684,  177265944,  2483900790,  14804568504,   40876316204,   51180241896,   23316504721;
  10: 1, 59019, 22465892, 1460063676, 31447182462, 291301853274, 1306296310676, 2941223165868, 3170078181769, 1291502224563;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+(6*n-7)*x+(14*n-21)*x^2+(10*n-17)*x^3+(2*n-4)*x^4)*P(n-1,x) + (2*x+4*x^2-4*x^4-2*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (6*n - 7)*x + (14*n - 21)*x^2 + (10*n - 17)*x^3 + (2*n - 4)*x^4)*P[n - 1, x] + (2*x + 4*x^2 - 4*x^4 - 2*x^5)*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; for(n = 2, N, my(i = n); v[i] = ((1 + (6*n - 7)*x + (14*n - 21)*x^2 + (10*n - 17)*x^3 + (2*n - 4)*x^4)*v[i - 1] + (2*x + 4*x^2 - 4*x^4 - 2*x^5)*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) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th …?
  - 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