A038333 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(361 + 1330\,x + 784\,x^{2} + 798\,n x + 784\,n x^{2} + 196\,n^{2} x^{2}\right)\,P_{n}(x) = \left(6859 + 45486\,x + 63308\,x^{2} + 24696\,x^{3} + 15162\,n x + 33516\,n x^{2} + 16464\,n x^{3} + 3724\,n^{2} x^{2} + 2744\,n^{2} x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 361 + 2128\,x + 1764\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	361	2128	1764															4253
1	6859	60648	100548	43904														211959
2	130321	1536416	3820824	3336704	960400													9784665
3	2476099	36489880	120992760	158493440	91238000	19361664												429051843
4	47045881	831969264	3448293660	6022750720	5200566000	2207229696	368947264											18126802485
5	893871739	18441985352	91724611356	200256461440	230558426000	146780774784	49069986112	6746464256										744472581039
6	16983563041	400454539072	2323690154352	6087796427776	8761220188000	7436892589056	3729318944512	1025462566912	119538913536									29901357886257
7	322687697779	8559715772664	56764430913456	173502198191616	299633730429600	317927158182144	212571179837184	87677049470976	20441154214656	2066104678400								1179465409388475
8	6131066257801	180705110756240	1348155234194580	4709345379486720	9488401463604000	12081232010921472	10097131042266240	5552879799828480	1941909650392320	392559888896000	34999813252096							45833450459855949
9	116490258898219	3776736814805416	31307160438518580	123031648039090560	283296557984748000	420829581713764608	422060077566728832	290137969541038080	135286372310664960	41022508389632000	7314960969688064	583137384431616						1758763201412008935
10	2213314919066161	78281453979603168	713803257998223624	3116801750323627520	8073951902565318000	13707020661534047232	16038282947535695616	13230291411071336448	7711323221707902720	3117710637612032000	833905550544439296	132955323650408448	9581271191425024					66766122704633125257
11	42052983462257059	1611293261080165208	16028127702323748648	76985003232993599744	221585124437070394000	423204262924863708288	565922269720188116736	544646996422436683776	380939367152370394368	192518631872542976000	68658223661492168704	16419982470825443328	2366573984281980928	155568095557812224				2511083477921489449011
12	799006685782884121	32969539034409534256	355290164068176428364	1861637350906936139264	5894164310026072480400	12508037104223749600512	18816915468196254881472	20696585864052593983488	16888311943755087483648	10241991215619286323200	4565771873489229218816	1455905112413189308416	314754339909503463424	41381113418378051584	2500201535750553600			93677014607334400334565
13	15181127029874798299	671165616057622661640	7789053596879252467980	44213887084039733307520	152712438941584605174000	356479057470376863614592	595868989826214737913280	737315871406873660661760	687595557710028561834240	486494582741916100352000	260248996788886065472512	103733239259439738224640	29901662291402829025280	5896808662118872350720	712557437688907776000	39825432462799929344		3469688875393000225563807
14	288441413567621167681	13602289818767819275904	169133735246520910733280	1033924744119082994268160	3868715119853476664408000	9851784860999506048985088	18114417290716928032563712	24904891656409954760130560	26128631192981085349701120	21127764736220356358144000	13185949170636893983940608	6306980946973936084058112	2272526334146615005921280	597543277761379064872960	108308730528713981952000	12106931468691178520576	629428514626908258304	127697198887810102766901345

Technical details

Node id: n-e64fc2aef53d02bf · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-33fb9c1876220e09

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:             361,             2128,              1764;
  1:            6859,            60648,            100548,              43904;
  2:          130321,          1536416,           3820824,            3336704,             960400;
  3:         2476099,         36489880,         120992760,          158493440,           91238000,           19361664;
  4:        47045881,        831969264,        3448293660,         6022750720,         5200566000,         2207229696,          368947264;
  5:       893871739,      18441985352,       91724611356,       200256461440,       230558426000,       146780774784,        49069986112,         6746464256;
  6:     16983563041,     400454539072,     2323690154352,      6087796427776,      8761220188000,      7436892589056,      3729318944512,      1025462566912,       119538913536;
  7:    322687697779,    8559715772664,    56764430913456,    173502198191616,    299633730429600,    317927158182144,    212571179837184,     87677049470976,     20441154214656,     2066104678400;
  8:   6131066257801,  180705110756240,  1348155234194580,   4709345379486720,   9488401463604000,  12081232010921472,  10097131042266240,   5552879799828480,   1941909650392320,   392559888896000,   34999813252096;
  9: 116490258898219, 3776736814805416, 31307160438518580, 123031648039090560, 283296557984748000, 420829581713764608, 422060077566728832, 290137969541038080, 135286372310664960, 41022508389632000, 7314960969688064, 583137384431616;

%F Formula

The row polynomials satisfy (361+(798*n+1330)*x+(196*n^2+784*n+784)*x^2)*P(n,x) = (6859+(15162*n+45486)*x+(3724*n^2+33516*n+63308)*x^2+(2744*n^2+16464*n+24696)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 2128*x + 1764*x^2.

%t Mathematica

P[0, x_] := 361 + 2128*x + 1764*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6859 + (15162*n + 45486)*x + (3724*n^2 + 33516*n + 63308)*x^2 + (2744*n^2 + 16464*n + 24696)*x^3)*P[n - 1, x])/(361 + (798*n + 1330)*x + (196*n^2 + 784*n + 784)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 361 + 2128*x + 1764*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6859 + (15162*n + 45486)*x + (3724*n^2 + 33516*n + 63308)*x^2 + (2744*n^2 + 16464*n + 24696)*x^3)*v[i - 1])/(361 + (798*n + 1330)*x + (196*n^2 + 784*n + 784)*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 (−∞, 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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^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

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.