Real-rooted atlas

A038284 → basis A007318 → θ+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(7 + 6\,x + 3\,n x\right)\,P_{n}(x) = \left(98 + 168\,x + 54\,x^{2} + 42\,n x + 18\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 196 + 336\,x + 108\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0196336108640
127447056453686415200
238416131712127008483846480352000
3537824230496029635201693440453600466568000000
47529536387233286223392047416320190512003919104326592179200000
5105413504632481024121978483211616998406223392001920360963200601622394883968000000
6147578905610119696384227693168642602207641617425497600716934758417923368962508226561511654487040000000
720661046784159385218048409847703552546463604736439122539520225834448896752781496321580182732819046845441007769601894400000000
8289254654976247932561408071723348121601092927209472010246192588800632336456908826347352371207374186086401333279180801410877440066512793640960000000000
94049565169664381816144568321227266178969602103884878233602254162369536001622996906065928114984530329628390616432640684416646144010863756288001024297021444353564672880640000000000
1056693912375296583137384431616206180718066892839272517727027204733740976025600389519257455820822721956684922889539247121367042874549913804806083703521280086040949800967313988648962829817036818841600000000000
117937147732541448844250330546176341135369892495367147598226318950495726761959628800886156310711992325907708738079948828935716268146688104633616862494722768085102182400521981762125824665572967055365150267006976182849716224401408000000000000
121111200682555801613334408190669619255718777082440908812735720475986493441876244534408724480192985152110611660814473886408295874568102000555081072643418031484174827521085089360055500802557710634416537643484100514283525047261666836483583854437990411754624614408519680000000000000
1315556809555781222420001612286004428809000725528701992960222875108329763635203581921383871201280040526881943228448768337724016193570406402126775145708781568010254094452524482560379781276019425280010742384664549457922282915276999884803533083166785536037630471598899202468471169024007522959753216180224000000000000000
1421779533378093711362986907434709994700814401160845923188736038403095589128503296066862532498929090560082527468684392477491275650179627359771033652933070293196341248028711464467068551168012153000832621608960040104902747651309568102274604409594839041978526573399900160280974187938447360276468770930688001685142984720384479588684267523801088000000000000000
Technical details
Node id
n-8fe4e03db036ba8f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5520686a41bcab10

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,            336,             108;
  1:          2744,           7056,            4536,             864;
  2:         38416,         131712,          127008,           48384,            6480;
  3:        537824,        2304960,         2963520,         1693440,          453600,           46656;
  4:       7529536,       38723328,        62233920,        47416320,        19051200,         3919104,         326592;
  5:     105413504,      632481024,      1219784832,      1161699840,       622339200,       192036096,       32006016,        2239488;
  6:    1475789056,    10119696384,     22769316864,     26022076416,     17425497600,      7169347584,     1792336896,      250822656,      15116544;
  7:   20661046784,   159385218048,    409847703552,    546463604736,    439122539520,    225834448896,    75278149632,    15801827328,    1904684544,     100776960;
  8:  289254654976,  2479325614080,   7172334812160,  10929272094720,  10246192588800,   6323364569088,  2634735237120,   737418608640,  133327918080,   14108774400,    665127936;
  9: 4049565169664, 38181614456832, 122726617896960, 210388487823360, 225416236953600, 162299690606592, 81149845303296, 28390616432640, 6844166461440, 1086375628800, 102429702144, 4353564672;
%F Formula
The row polynomials satisfy (7+(3*n+6)*x)*P(n,x) = (98+(42*n+168)*x+(18*n+54)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 336*x + 108*x^2.
%t Mathematica
P[0, x_] := 196 + 336*x + 108*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((98 + (42*n + 168)*x + (18*n + 54)*x^2)*P[n - 1, x])/(7 + (3*n + 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 196 + 336*x + 108*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((98 + (42*n + 168)*x + (18*n + 54)*x^2)*v[i - 1])/(7 + (3*n + 6)*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

Proof chains ? JSON

Same fingerprint ?

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.