A038284 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(7 + 24\,x + 12\,n x\right)\,P_{n}(x) = \left(98 + 672\,x + 864\,x^{2} + 168\,n x + 288\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 196 + 1344\,x + 1728\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	196	1344	1728															3268
1	2744	28224	72576	55296														158840
2	38416	526848	2032128	3096576	1658880													7352848
3	537824	9219840	47416320	108380160	116121600	47775744												329451488
4	7529536	154893312	995742720	3034644480	4877107200	4013162496	1337720832											14420800576
5	105413504	2529924096	19516557312	74348789760	159318835200	196644962304	131096641536	36691771392										620252895104
6	1475789056	40478785536	364309069824	1665412890624	4460927385600	7341411926016	7341411926016	4109478395904	990677827584									26315583996160
7	20661046784	637540872192	6557563256832	34973670703104	112415370117120	231254475669504	308339300892672	258897138941952	124825406275584	26418075402240								1104339203177984
8	289254654976	9917302456320	114757356994560	699473414062080	2623025302732800	6475125318746112	10791875531243520	12081866483957760	8737778439290880	3698530556313600	697437190619136							45930076151071744
9	4049565169664	152726457827328	1963625886351360	13464863220695040	57706556660121600	166194883181150208	332389766362300416	465151859632373760	448539293216931840	284786852836147200	107405327355346944	18260173718028288						1896019978092443648
10	56693912375296	2332549537726464	32988914890702848	251344113452974080	1211837689862553600	3988677196347604992	9306913458144411648	15629102483647758336	18838650315111137280	15948063758824243200	9022047497849143296	3067709184628752384	474764516668735488					77774488372878118912
11	793714773254144	35377001322184704	545816591827992576	4574462864844128256	24506051061664972800	90742406216908013568	241979749911754702848	474082775337315336192	685726871470045396992	725636901026503065600	547337548202848026624	279161535801216466944	86407142033709858816	12270836738515009536				3173008267973248409600
12	11112006825558016	533376327626784768	8915004333190545408	81508611046313558016	480318600808633466880	1976167957612663406592	5928503872837990219776	13274317709444829413376	22400411134688149635072	28444966520238920171520	26819539861939553304576	18238553672346142507008	8467899919303566163968	2405084000748941869056	315535801847528816640			128842267155530875420672
13	155568095557812224	8000644914401771520	144011608459231887360	1426400693310487265280	9169718742710275276800	41499527109865931538432	138331757032886438461440	348450839872926772101120	672012334040644489052160	995573828208362206003200	1126420674201461238792192	957524067798172481617920	592752994351249631477760	252533820078638896250880	66262518387981051494400	8077716527296737705984		5210188364866975828508672
14	2177953337809371136	119476297388399788032	2304185735347710197760	24577981177042242109440	171168083197258471833600	845081279328178969509888	3098631357536656221536256	8672554236837288550072320	18816345353138045693460480	31858362502667590592102400	42053038503521219581575168	42897078237358127176482816	33194167683669979362754560	18855858565871704253399040	7421402059453877767372800	1809408502114469246140416	205981771446066811502592	209926081957303578859208704

Technical details

Node id: n-80f979e5986f600d · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-f565be5c3ba48d91

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,            1344,             1728;
  1:          2744,           28224,            72576,             55296;
  2:         38416,          526848,          2032128,           3096576,           1658880;
  3:        537824,         9219840,         47416320,         108380160,         116121600,           47775744;
  4:       7529536,       154893312,        995742720,        3034644480,        4877107200,         4013162496,         1337720832;
  5:     105413504,      2529924096,      19516557312,       74348789760,      159318835200,       196644962304,       131096641536,        36691771392;
  6:    1475789056,     40478785536,     364309069824,     1665412890624,     4460927385600,      7341411926016,      7341411926016,      4109478395904,       990677827584;
  7:   20661046784,    637540872192,    6557563256832,    34973670703104,   112415370117120,    231254475669504,    308339300892672,    258897138941952,    124825406275584,     26418075402240;
  8:  289254654976,   9917302456320,  114757356994560,   699473414062080,  2623025302732800,   6475125318746112,  10791875531243520,  12081866483957760,   8737778439290880,   3698530556313600,    697437190619136;
  9: 4049565169664, 152726457827328, 1963625886351360, 13464863220695040, 57706556660121600, 166194883181150208, 332389766362300416, 465151859632373760, 448539293216931840, 284786852836147200, 107405327355346944, 18260173718028288;

%F Formula

The row polynomials satisfy (7+(12*n+24)*x)*P(n,x) = (98+(168*n+672)*x+(288*n+864)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 1344*x + 1728*x^2.

%t Mathematica

P[0, x_] := 196 + 1344*x + 1728*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((98 + (168*n + 672)*x + (288*n + 864)*x^2)*P[n - 1, x])/(7 + (12*n + 24)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 196 + 1344*x + 1728*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((98 + (168*n + 672)*x + (288*n + 864)*x^2)*v[i - 1])/(7 + (12*n + 24)*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 (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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 ?

A038271 → x ↦ x/(x+1) → θ+1

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.