A257608 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(9\,x + 27\,x^{2} + 27\,x^{3} + 9\,x^{4}\right)\,P_{n-1}'(x)\)

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

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	2														3
2	1	22	22													45
3	1	222	660	440												1323
4	1	2222	14982	25520	12760											55485
5	1	22222	309100	1014200	1212200	484880										3042603
6	1	222222	6117342	34579600	74263200	68368080	22789360									206339805
7	1	2222222	118673940	1090575640	3750546800	5966933280	4466714560	1276204160								16671870603
8	1	22222222	2279249302	32909596720	170396925160	416999709280	526115164960	331813081600	82953270400							1563489219645
9	1	222222222	43550181180	967053694200	7259064535800	25657069782960	48535320408960	50698486460160	27623439043200	6138542009600						166922748338283
10	1	2222222222	829897886862	27948507061200	296629944956400	1456069662376560	3875324902291920	5962681076352000	5311912670064000	2547494933984000	509498986796800					19988392803991965
11	1	22222222222	15792504294820	799156155450840	11785814668161600	78251142377664960	281578143757753920	598629783414875520	775343894877552000	601976122171424000	257806487319180800	46873906785305600				2653060266253886283
12	1	222222222222	300302026046022	22692222438519920	459250671230053560	4047413506762728960	19164601598426718720	54080682188986252800	95511040248028204800	106737490238995616000	73424408486273644800	28405587511895193600	4734264585315865600			386587731782601067005
13	1	2222222222222	5708182939318860	641388268799478200	17650349286229059400	203632546817827567440	1244281522731317790720	4534381349607016427520	10487550280390165382400	15820291177951727667200	15540832862087537804800	9595604331926694681600	3384999178500843904000	520769104384745216000		61350639802358066520363
14	1	22222222222222	108479920291502782	18073035185171766800	671663183385660065600	10037810426496772359440	78006213450660374159280	359880171647802847641600	1060325958192938140704000	2073342597340913907008000	2727299958219959336614400	2389317056504722946201600	1338120947981188148377600	433800663952492764928000	61971523421784680704000	10532792745859673264255325

Technical details

Node id: n-d60e41622fb1fd8b · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-c5a9a6448f2c45fc

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,        22,          22;
  3: 1,       222,         660,          440;
  4: 1,      2222,       14982,        25520,         12760;
  5: 1,     22222,      309100,      1014200,       1212200,         484880;
  6: 1,    222222,     6117342,     34579600,      74263200,       68368080,       22789360;
  7: 1,   2222222,   118673940,   1090575640,    3750546800,     5966933280,     4466714560,     1276204160;
  8: 1,  22222222,  2279249302,  32909596720,  170396925160,   416999709280,   526115164960,   331813081600,    82953270400;
  9: 1, 222222222, 43550181180, 967053694200, 7259064535800, 25657069782960, 48535320408960, 50698486460160, 27623439043200, 6138542009600;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (9*x+27*x^2+27*x^3+9*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 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (9*x + 27*x^2 + 27*x^3 + 9*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 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (9*x + 27*x^2 + 27*x^3 + 9*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) = 9*x + 1.?
  - 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) = 9*x + 1.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257608 → 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