A257624 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(5 + 20\,x + 25\,x^{2} + 10\,x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	10														15
2	25	130	130													285
3	125	1290	3120	2080												6615
4	625	11570	51090	79040	39520											181845
5	3125	98810	712400	1924000	2173600	869440										5781375
6	15625	821730	9120930	38334400	73507200	65208000	21736000									208743885
7	78125	6730090	111012720	682616480	1977976000	2921318400	2130128000	608608000								8438467815
8	390625	54621970	1308631090	11332834240	46595305120	101941840000	122025904000	75467392000	18866848000							377593767045
9	1953125	440882010	15105027600	179597776800	1007444037600	3063933512640	5355141792000	5378877504000	2886627744000	641472832000						18528643061775
10	9765625	3546587330	171886769730	2756049316800	20538906398400	83442439080000	199766599032000	289794785280000	250457407200000	118672473920000	23734494784000					989338598133885
11	48828125	28470354890	1936860102320	41334878750880	401526345792000	2120704722364800	6680692754736000	13128129190176000	16246903152480000	12313071010240000	5221588852480000	949379791360000				57105296077665015
12	244140625	228251120370	21675575739090	609678064149440	7611310574730720	51247674054720000	206673551418048000	528390756077184000	878132196317376000	946412979512000000	638339237215680000	244939986170880000	40823331028480000			3543202604504248245
13	1220703125	1828450369210	241398597694800	8882302109917600	140976162989261600	1192402313738475840	6034683533983104000	19525019097712128000	41845947131596608000	60141710019174784000	57359153544492800000	34841288963120640000	12206175977515520000	1877873227310080000		235174355502012086175
14	6103515625	14639809984930	2679154429442530	128214607101963200	2565358510905881600	26949521860533272000	168607779125073288000	676621635492042240000	1818808058845377600000	3347296923087877760000	4232813644766715008000	3618335122378296320000	1998628640492323840000	644110516967357440000	92015788138193920000	16626883898072141475885

Technical details

Node id: n-96937d0c31038dc0 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-1d32ce7fdc01b2c4

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:       5,        10;
  2:      25,       130,         130;
  3:     125,      1290,        3120,         2080;
  4:     625,     11570,       51090,        79040,         39520;
  5:    3125,     98810,      712400,      1924000,       2173600,        869440;
  6:   15625,    821730,     9120930,     38334400,      73507200,      65208000,      21736000;
  7:   78125,   6730090,   111012720,    682616480,    1977976000,    2921318400,    2130128000,     608608000;
  8:  390625,  54621970,  1308631090,  11332834240,   46595305120,  101941840000,  122025904000,   75467392000,   18866848000;
  9: 1953125, 440882010, 15105027600, 179597776800, 1007444037600, 3063933512640, 5355141792000, 5378877504000, 2886627744000, 641472832000;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (5+20*x+25*x^2+10*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[((5 + 20*x + 25*x^2 + 10*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] = ((5 + 20*x + 25*x^2 + 10*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 + 5.?
  - 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 + 5.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257624 → 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