A038328 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(14 + 36\,x + 22\,x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 196 + 616\,x + 484\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	196	616	484															1296
1	2744	12936	20328	10648														46656
2	38416	241472	569184	596288	234256													1679616
3	537824	4225760	13280960	20870080	16397920	5153632												60466176
4	7529536	70992768	278900160	584362240	688712640	432905088	113379904											2176782336
5	105413504	1159548544	5466443136	14316874880	22497946240	21212349312	11111230592	2494357888										78364164096
6	1475789056	18552776704	102040271872	320697997312	629942494720	791927707648	622228913152	279368083456	54875873536									2821109907456
7	20661046784	292206233088	1836724893696	6734657943552	15874550866944	24945722790912	26133614352384	17600189257728	6914360065536	1207269217792								101559956668416
8	289254654976	4545430292480	32142685639680	134693158871040	370406186895360	698480238145536	914676502333440	821342165360640	484005204587520	169017690490880	26559922791424							3656158440062976
9	4049565169664	69999626504192	549997065390080	2592843308267520	8148936111697920	17927659445735424	28172036271869952	31621673366384640	24845600502159360	13014362167797760	4090228109879296	584318301411328						131621703842267136
10	56693912375296	1069085204791296	9239950698553344	48399741754327040	171127658345656320	430263826697650176	788817015612358656	1062488225110523904	1043515221090693120	728804281396674560	343579161229860864	98165474637103104	12855002631049216					4738381338321616896
11	793714773254144	16214458939334656	152879184285155328	880875299928752128	3460581535434383360	9788502057371541504	20509242405921325056	32228809495019225088	37983954047701229568	33160594803548692480	20843802447944892416	8933058191976382464	2339610478850957312	282810057883082752				170581728179578208256
12	11112006825558016	244464150162276352	2497026676657537024	15695596253275947008	67827398094513913856	213171822582758014976	502476438945072463872	902406665860538302464	1240809165558240165888	1299895316299108745216	1021346319949299728384	583626468542456987648	229281826927393816576	55430771345084219392	6221821273427820544			6140942214464815497216
13	155568095557812224	3666962252434145280	40336584776775598080	274672934432329072640	1294886690895265628160	4476608274237918314496	11724450242051690823680	23688174978839130439680	37224274966747204976640	45496336070468806082560	42896545437870588592128	30640389598478991851520	16049727884917567160320	5820230991233843036160	1306582467419842314240	136880068015412051968		221073919720733357899776
14	2177953337809371136	54759969636349902848	645385356428409569280	4732825947141670174720	24171218230044958392320	91160023039026700222464	262627685421957874450432	589572355028885024276480	1042279699068921739345920	1455882754255001794641920	1601471029680501974106112	1372689454011858834948096	898784761555383760977920	434577247345460280033280	146337236351022339194880	30661135235452299640832	3011361496339065143296	7958661109946400884391936

Technical details

Node id: n-89a876c443c480e3 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-428cb055b0457fc7

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:           196,            616,             484;
  1:          2744,          12936,           20328,            10648;
  2:         38416,         241472,          569184,           596288,           234256;
  3:        537824,        4225760,        13280960,         20870080,         16397920,           5153632;
  4:       7529536,       70992768,       278900160,        584362240,        688712640,         432905088,         113379904;
  5:     105413504,     1159548544,      5466443136,      14316874880,      22497946240,       21212349312,       11111230592,        2494357888;
  6:    1475789056,    18552776704,    102040271872,     320697997312,     629942494720,      791927707648,      622228913152,      279368083456,       54875873536;
  7:   20661046784,   292206233088,   1836724893696,    6734657943552,   15874550866944,    24945722790912,    26133614352384,    17600189257728,     6914360065536,     1207269217792;
  8:  289254654976,  4545430292480,  32142685639680,  134693158871040,  370406186895360,   698480238145536,   914676502333440,   821342165360640,   484005204587520,   169017690490880,   26559922791424;
  9: 4049565169664, 69999626504192, 549997065390080, 2592843308267520, 8148936111697920, 17927659445735424, 28172036271869952, 31621673366384640, 24845600502159360, 13014362167797760, 4090228109879296, 584318301411328;

%F Formula

The row polynomials satisfy (1+x)*P(n,x) = (14+36*x+22*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 616*x + 484*x^2.

%t Mathematica

P[0, x_] := 196 + 616*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14 + 36*x + 22*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 196 + 616*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14 + 36*x + 22*x^2)*v[i - 1])/(1 + x)); 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 basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.