A038328 → 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

\(\left(7 + 8\,x + 4\,n x\right)\,P_{n}(x) = \left(98 + 224\,x + 96\,x^{2} + 56\,n x + 32\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 196 + 448\,x + 192\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	196	448	192															836
1	2744	9408	8064	2048														22264
2	38416	175616	225792	114688	20480													574992
3	537824	3073280	5268480	4014080	1433600	196608												14523872
4	7529536	51631104	110638080	112394240	60211200	16515072	1835008											360754240
5	105413504	843308032	2168506368	2753658880	1966899200	809238528	179830784	16777216										8843632512
6	1475789056	13492928512	40478785536	61681958912	55073177600	30211571712	10070523904	1879048192	150994944									214514778368
7	20661046784	212513624064	728618139648	1295321137152	1387844075520	951664508928	422962003968	118380036096	19025362944	1342177280								5158332112384
8	289254654976	3305767485440	12750817443840	25906422743040	32383028428800	26646606249984	14803670138880	5524401684480	1331775406080	187904819200	11811160064							123141460214784
9	4049565169664	50908819275776	218180654039040	498698637803520	712426625433600	683929560416256	455953040277504	212689464852480	68364470845440	14468671078400	1818918649856	103079215104						2921591507056640
10	56693912375296	777516512575488	3665434987855872	9309041238999040	14960959134105600	16414309449990144	12766685127770112	7146366019043328	2871307775508480	810245580390400	152789166587904	17317308137472	893353197568					68949559566536704
11	793714773254144	11792333774061568	60646287980888064	169424550549782528	302543840267468800	373425539987275776	331933813322022912	216773102577647616	104515603028508672	36866173907763200	9269209439666176	1575875040509952	162590281957376	7696581394432				1619730331512201216
12	11112006825558016	177792109208928256	990556037021171712	3018837446159761408	5929859269242388480	8132378426389561344	8132378426389561344	6069646872174133248	3414176365597949952	1445154017184317440	454191262543642624	102957169313316864	15933847631822848	1508529953308672	65970697666560			37896547756333088768
13	155568095557812224	2666881638133923840	16001289828803543040	52829655307795824640	113206404230991052800	170779946954180788224	189755496615756431360	159328230394570997760	102425290967938498560	50580390601451110400	19076033026832990208	5405251388949135360	1115369334227599360	158395645097410560	13853846509977600	562949953421312		883498620826750517248
14	2177953337809371136	39825432462799929344	256020637260856688640	910295599149712670720	2113186212311832985600	3477700737976045142016	4250523124192944062464	3965502623153767055360	2867908147102277959680	1618572499246435532800	712171899668431634432	242155262224921264128	62460682716745564160	11826874833939988480	1551630809117491200	126100789566373888	4785074604081152	20532010202311807795200

Technical details

Node id: n-f37732e25746403d · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-604d756e1453b62f

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,            448,             192;
  1:          2744,           9408,            8064,            2048;
  2:         38416,         175616,          225792,          114688,           20480;
  3:        537824,        3073280,         5268480,         4014080,         1433600,          196608;
  4:       7529536,       51631104,       110638080,       112394240,        60211200,        16515072,         1835008;
  5:     105413504,      843308032,      2168506368,      2753658880,      1966899200,       809238528,       179830784,        16777216;
  6:    1475789056,    13492928512,     40478785536,     61681958912,     55073177600,     30211571712,     10070523904,      1879048192,      150994944;
  7:   20661046784,   212513624064,    728618139648,   1295321137152,   1387844075520,    951664508928,    422962003968,    118380036096,    19025362944,     1342177280;
  8:  289254654976,  3305767485440,  12750817443840,  25906422743040,  32383028428800,  26646606249984,  14803670138880,   5524401684480,  1331775406080,   187904819200,   11811160064;
  9: 4049565169664, 50908819275776, 218180654039040, 498698637803520, 712426625433600, 683929560416256, 455953040277504, 212689464852480, 68364470845440, 14468671078400, 1818918649856, 103079215104;

%F Formula

The row polynomials satisfy (7+(4*n+8)*x)*P(n,x) = (98+(56*n+224)*x+(32*n+96)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 448*x + 192*x^2.

%t Mathematica

P[0, x_] := 196 + 448*x + 192*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((98 + (56*n + 224)*x + (32*n + 96)*x^2)*P[n - 1, x])/(7 + (4*n + 8)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 196 + 448*x + 192*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((98 + (56*n + 224)*x + (32*n + 96)*x^2)*v[i - 1])/(7 + (4*n + 8)*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)*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 ?

A038255 → basis A013611 → 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.