Real-rooted atlas

A038338 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + 6\,x + 3\,n x\right)\,P_{n}(x) = \left(48 + 288\,x + 324\,x^{2} + 72\,n x + 108\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 576 + 3456\,x + 3888\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0576345638887920
113824124416279936186624604800
233177639813121343692817915904839808044064000
37962624119439360537477120107495424010077696003627970563110400000
419110297634398535681934917632051597803520725594112005224277606415237476352214617600000
5458647142496315899904650132324352216710774784040633270272004388393189376255989602713662691331276814556672000000
61100753141762641807540224208042343792648321693751705619503969730560028085716412006424575001860505612036735605145625389989167104974177280000000
7264180754022471328803586048641959232274432299580975061401684257149236019201516628686248345617694001339564032129996744535572485484237660094464101559956668416064497254400000000

All 15 rows

Technical details
Node id
n-829df31fc8588be3 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-311883d116e971c2

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:              576,              3456,               3888;
  1:            13824,            124416,             279936,              186624;
  2:           331776,           3981312,           13436928,            17915904,              8398080;
  3:          7962624,         119439360,          537477120,          1074954240,           1007769600,            362797056;
  4:        191102976,        3439853568,        19349176320,         51597803520,          72559411200,          52242776064,          15237476352;
  5:       4586471424,       96315899904,       650132324352,       2167107747840,        4063327027200,        4388393189376,        2559896027136,         626913312768;
  6:     110075314176,     2641807540224,     20804234379264,      83216937517056,      195039697305600,      280857164120064,      245750018605056,      120367356051456,       25389989167104;
  7:    2641807540224,    71328803586048,    641959232274432,    2995809750614016,     8425714923601920,    15166286862483456,    17694001339564032,    12999674453557248,     5484237660094464,     1015599566684160;
  8:   63403380965376,  1902101428961280,  19258776968232960,  102713477163909120,   337028596944076800,   727981769399205888,  1061640080373841920,  1039973956284579840,   658108519211335680,   243743896004198400,    40217742840692736;
  9: 1521681143169024, 50215477724577792, 564924124401500160, 3389544746409000960, 12710792799033753600, 32031197853565059072, 56054596243738853376, 68638281114782269440, 57913549690597539840, 32174194272554188800, 10617484109942882304, 1579460446107205632;
%F Formula
The row polynomials satisfy (2+(3*n+6)*x)*P(n,x) = (48+(72*n+288)*x+(108*n+324)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 3456*x + 3888*x^2.
%t Mathematica
P[0, x_] := 576 + 3456*x + 3888*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((48 + (72*n + 288)*x + (108*n + 324)*x^2)*P[n - 1, x])/(2 + (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] = 576 + 3456*x + 3888*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((48 + (72*n + 288)*x + (108*n + 324)*x^2)*v[i - 1])/(2 + (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.