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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	65536	15198040	432842344	4053124768	17335369744	38710376320	46803137920	29056867840	7264216960							143671199472
9	262144	106910568	4495601880	58750414800	346269036960	1082204544960	1920007689600	1943303281920	1046047242240	232454942720						6633639927792
10	1048576	750471128	46132035048	826693818720	6539061642960	27487267093440	67130873705280	98502781977600	85645117958400	40679614976000	8135922995200					334994217722352
11	4194304	5261686504	469575532888	11392868134032	118678781205600	653039307634560	2109086364665280	4207972265777280	5254643980185600	4001711838924800	1700407905996800	309165073817600				18366573227755248
12	16777216	36865359960	4753633880424	154681343202848	2092539257568144	14781322469168640	61419804098231040	160035552125729280	269161227152737920	292201674372608000	197873783166259200	76054608159129600	12675768026521600			1086455750687173872
13	67108864	258191737448	47941857763800	2077408335962960	36110210955538720	322695912065567040	1691206106951961600	5597803709697239040	12177545371922810880	17671411174438461440	16954767230694092800	10333224262205644800	3625269655585177600	557733793166950400		68969893036136017392
14	268435456	1807879033048	482258686749928	27677494376211680	613079316999989840	6853426548356548160	44628540330451197120	183916451523181977600	503307577995058636800	937495198309122252800	1194961476948945305600	1026610452467984384000	568672980973838540800	183494417951926681600	26213488278846668800	4676795252205922613232

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