A257622 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(4 + 16\,x + 20\,x^{2} + 8\,x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	8							12
2	16	88	88						192
3	64	744	1848	1232					3888
4	256	5720	26664	41888	20944				95472
5	1024	42088	329560	917840	1047200	418880			2756592
6	4096	302808	3758568	16545760	32358480	28902720	9634240		91506672
7	16384	2152424	40916568	267714832	799013600	1196321280	876715840	250490240	3433341168

All 15 rows

Technical details

Node id: n-c7bafe77bda3d4ca · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-31113a40d89e3fd5

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:      4,         8;
  2:     16,        88,         88;
  3:     64,       744,       1848,        1232;
  4:    256,      5720,      26664,       41888,        20944;
  5:   1024,     42088,     329560,      917840,      1047200,        418880;
  6:   4096,    302808,    3758568,    16545760,     32358480,      28902720,       9634240;
  7:  16384,   2152424,   40916568,   267714832,    799013600,    1196321280,     876715840,     250490240;
  8:  65536,  15198040,  432842344,  4053124768,  17335369744,   38710376320,   46803137920,   29056867840,    7264216960;
  9: 262144, 106910568, 4495601880, 58750414800, 346269036960, 1082204544960, 1920007689600, 1943303281920, 1046047242240, 232454942720;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (4+16*x+20*x^2+8*x^3)*P(n-1,x) + (3*x+9*x^2+9*x^3+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[((4 + 16*x + 20*x^2 + 8*x^3)*P[n - 1, x] + (3*x + 9*x^2 + 9*x^3 + 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] = ((4 + 16*x + 20*x^2 + 8*x^3)*v[i - 1] + (3*x + 9*x^2 + 9*x^3 + 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 read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257622 → 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