A257614 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 10\,x^{2} + 4\,x^{3}\right)\,P_{n-1}(x) + \left(5\,x + 15\,x^{2} + 15\,x^{3} + 5\,x^{4}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	4														6
2	4	36	36													76
3	8	268	756	504												1536
4	16	1908	11484	19152	9576											42136
5	32	13420	154980	486360	574560	229824										1459176
6	64	94068	1980540	10437840	21881160	19994688	6664896									61053256
7	128	658732	24613092	205170840	679704480	1065004416	793122624	226606464								2994880776
8	256	4611636	301285692	3832487568	18851744520	45068026752	56265052032	35350608384	8837652096							168511468936
9	512	32282476	3656933028	69370288344	487555643232	1669278590784	3107454440832	3220984279296	1749855115008	388856692224						10697044265736
10	1024	225979380	44173738620	1230491964240	12044259629640	56771857388736	147847621239360	224829869322240	199112301722880	95269889594880	19053977918976					756204668499976
11	2048	1581859756	532118677860	21536795732760	288353059172640	1821902380607232	6377507743932864	13345524287061120	17131081575928320	13238626086766080	5659031441935872	1028914807624704				58913011879301256
12	4096	11073026484	6399660872124	373575188946960	6752966420720520	56111837696538624	256915416843461376	710619661914978816	1239980873384373120	1375983015418851840	942962313232203264	364235841899145216	60705973649857536			5014647886382979976
13	8192	77511201772	76895587703844	6440373464308056	155663189845843680	1677090811903835328	9848536632190384128	35028051663531902976	79793363496827844864	119230360153598453760	116457208043598309888	71681407902790253568	25253685038340734976	3885182313590882304		463017066592781667336
14	16384	542578445172	923444653262076	110562887121090768	3546957272430414024	49017368477703802176	363788805775303516608	1630888157045153470464	4717415281744513700352	9114734921079552500736	11898062465793436348416	10374529484813353187328	5794931538641750529024	1876543057464396152832	268077579637770878976	46091647104619717315336

Technical details

Node id: n-9bf05be8e4114bb6 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-dcf8348fea013f35

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:   2,        4;
  2:   4,       36,         36;
  3:   8,      268,        756,         504;
  4:  16,     1908,      11484,       19152,         9576;
  5:  32,    13420,     154980,      486360,       574560,        229824;
  6:  64,    94068,    1980540,    10437840,     21881160,      19994688,       6664896;
  7: 128,   658732,   24613092,   205170840,    679704480,    1065004416,     793122624,     226606464;
  8: 256,  4611636,  301285692,  3832487568,  18851744520,   45068026752,   56265052032,   35350608384,    8837652096;
  9: 512, 32282476, 3656933028, 69370288344, 487555643232, 1669278590784, 3107454440832, 3220984279296, 1749855115008, 388856692224;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (2+8*x+10*x^2+4*x^3)*P(n-1,x) + (5*x+15*x^2+15*x^3+5*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[((2 + 8*x + 10*x^2 + 4*x^3)*P[n - 1, x] + (5*x + 15*x^2 + 15*x^3 + 5*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] = ((2 + 8*x + 10*x^2 + 4*x^3)*v[i - 1] + (5*x + 15*x^2 + 15*x^3 + 5*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), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - 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), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A257614 → 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