A167884 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(8\,x + 24\,x^{2} + 24\,x^{3} + 8\,x^{4}\right)\,P_{n-1}'(x)\)

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

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	20	20													41
3	1	182	540	360												1083
4	1	1640	11000	18720	9360											40721
5	1	14762	203400	666000	795600	318240										1998003
6	1	132860	3605420	20311200	43570800	40098240	13366080									121084601
7	1	1195742	62620740	572677560	1965927600	3125435040	2339064000	668304000								8735224683
8	1	10761680	1076510000	15444112320	79765227360	194984375040	245882407680	155046528000	38761632000							730971554081
9	1	96855122	18408286800	405479988000	3033799423200	10706377106880	20237581728000	21131772480000	11512204704000	2558267712000						69603988284003
10	1	871696100	313909426820	10468348862400	110657860653600	542110641770880	1441309343160960	2216390117760000	1973936109600000	946559053440000	189311810688000					7431058067058761
11	1	7845264902	5345177216940	267359091242760	3923886471991200	25988903574828480	93392804769264000	198399703870368000	256856473954080000	199378594134720000	85379626620288000	15523568476416000				879116273985680283
12	1	70607384120	90946465337000	6780190470973920	136439589948021360	1198957186615668480	5667781383836732160	15978423359074176000	28202853631496544000	31507164652803840000	21669765722212608000	8382726977264640000	1397121162877440000			114148104873673365041
13	1	635466457082	1546795984570200	171141798150414000	4678791420530026800	53796190709475133920	328077489645857952000	1194159200659064640000	2759934040873577568000	4161411359333452224000	4086781207806705408000	2522983490197986816000	889966180752929280000	136917873961989120000		16138878514591169610003
14	1	5719198113740	26301886402264220	4306387281482613600	158849803629401648400	2364722727386501401920	18335237002444995272640	84470948919859582080000	248656946295008291040000	485938675928998131840000	638973718423018602624000	559661589667770670080000	313395233650852953600000	101593062479795927040000	14513294639970846720000	2468066612233622986338521

Technical details

Node id: n-56a2efc225b761f1 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-b84a4525108a87c3

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,       20,          20;
  3: 1,      182,         540,          360;
  4: 1,     1640,       11000,        18720,          9360;
  5: 1,    14762,      203400,       666000,        795600,         318240;
  6: 1,   132860,     3605420,     20311200,      43570800,       40098240,       13366080;
  7: 1,  1195742,    62620740,    572677560,    1965927600,     3125435040,     2339064000,      668304000;
  8: 1, 10761680,  1076510000,  15444112320,   79765227360,   194984375040,   245882407680,   155046528000,    38761632000;
  9: 1, 96855122, 18408286800, 405479988000, 3033799423200, 10706377106880, 20237581728000, 21131772480000, 11512204704000, 2558267712000;

%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) + (8*x+24*x^2+24*x^3+8*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] + (8*x + 24*x^2 + 24*x^3 + 8*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] + (8*x + 24*x^2 + 24*x^3 + 8*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) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n…?
  - 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) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n…?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A167884 → 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