A209414 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-3 - 9\,x - 9\,x^{2} - 3\,x^{3} + 4\,n + 12\,n x + 12\,n x^{2} + 4\,n x^{3}\right)\,P_{n}(x) = \left(3 + 9\,x + 9\,x^{2} + 3\,x^{3} - 6\,n - 20\,n x - 24\,n x^{2} - 12\,n x^{3} - 2\,n x^{4} + 4\,n^{2} + 16\,n^{2} x + 24\,n^{2} x^{2} + 16\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(1 + 27\,x + 116\,x^{2} + 214\,x^{3} + 201\,x^{4} + 95\,x^{5} + 18\,x^{6} - n - 14\,n x - 55\,n x^{2} - 100\,n x^{3} - 95\,n x^{4} - 46\,n x^{5} - 9\,n x^{6}\right)\,P_{n-1}'(x) + \left(x + 14\,x^{2} + 55\,x^{3} + 100\,x^{4} + 95\,x^{5} + 46\,x^{6} + 9\,x^{7}\right)\,P_{n-1}''(x)\)

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

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	2														3
2	1	6	6													13
3	1	10	26	18												55
4	1	14	62	102	54											233
5	1	18	114	314	378	162										987
6	1	22	182	718	1422	1350	486									4181
7	1	26	266	1378	3890	5994	4698	1458								17711
8	1	30	366	2358	8742	19086	24030	16038	4374							75025
9	1	34	482	3722	17194	49202	87282	92826	53946	13122						317811
10	1	38	614	5534	30718	109526	253974	378702	348462	179334	39366					1346269
11	1	42	762	7858	51042	218874	631754	1227906	1577394	1278666	590490	118098				5702887
12	1	46	926	10758	80150	402718	1397902	3377142	5639814	6359310	4605822	1928934	354294			24157817
13	1	50	1106	14298	120282	694210	2824930	8202602	16999146	24856146	24962418	16336890	6259194	1062882		102334155
14	1	54	1302	18542	173934	1135206	5310278	18075294	44984094	81493398	105890166	95829966	57198798	20194758	3188646	433494437

Technical details

Node id: n-41688b0638635402 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-23b2bc8195b15215

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,  2;
  2: 1,  6,   6;
  3: 1, 10,  26,   18;
  4: 1, 14,  62,  102,    54;
  5: 1, 18, 114,  314,   378,   162;
  6: 1, 22, 182,  718,  1422,  1350,   486;
  7: 1, 26, 266, 1378,  3890,  5994,  4698,  1458;
  8: 1, 30, 366, 2358,  8742, 19086, 24030, 16038,  4374;
  9: 1, 34, 482, 3722, 17194, 49202, 87282, 92826, 53946, 13122;

%F Formula

The row polynomials satisfy (4*n-3+(12*n-9)*x+(12*n-9)*x^2+(4*n-3)*x^3)*P(n,x) = (4*n^2-6*n+3+(16*n^2-20*n+9)*x+(24*n^2-24*n+9)*x^2+(16*n^2-12*n+3)*x^3+(4*n^2-2*n)*x^4)*P(n-1,x) - (n-1+(14*n-27)*x+(55*n-116)*x^2+(100*n-214)*x^3+(95*n-201)*x^4+(46*n-95)*x^5+(9*n-18)*x^6)*d/dx P(n-1,x) + (x+14*x^2+55*x^3+100*x^4+95*x^5+46*x^6+9*x^7)*d^2/dx^2 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[((4*n^2 - 6*n + 3 + (16*n^2 - 20*n + 9)*x + (24*n^2 - 24*n + 9)*x^2 + (16*n^2 - 12*n + 3)*x^3 + (4*n^2 - 2*n)*x^4)*P[n - 1, x] - (n - 1 + (14*n - 27)*x + (55*n - 116)*x^2 + (100*n - 214)*x^3 + (95*n - 201)*x^4 + (46*n - 95)*x^5 + (9*n - 18)*x^6)*D[P[n - 1, x], x] + (x + 14*x^2 + 55*x^3 + 100*x^4 + 95*x^5 + 46*x^6 + 9*x^7)*D[P[n - 1, x], {x, 2}])/(4*n - 3 + (12*n - 9)*x + (12*n - 9)*x^2 + (4*n - 3)*x^3)]];
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] = ((4*n^2 - 6*n + 3 + (16*n^2 - 20*n + 9)*x + (24*n^2 - 24*n + 9)*x^2 + (16*n^2 - 12*n + 3)*x^3 + (4*n^2 - 2*n)*x^4)*v[i - 1] - (n - 1 + (14*n - 27)*x + (55*n - 116)*x^2 + (100*n - 214)*x^3 + (95*n - 201)*x^4 + (46*n - 95)*x^5 + (9*n - 18)*x^6)*deriv(v[i - 1], x) + (x + 14*x^2 + 55*x^3 + 100*x^4 + 95*x^5 + 46*x^6 + 9*x^7)*deriv(deriv(v[i - 1], x), x))/(4*n - 3 + (12*n - 9)*x + (12*n - 9)*x^2 + (4*n - 3)*x^3)); 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 coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? 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