Real-rooted atlas

A075505 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(11 + x\right)\,P_{n-1}(x) + \left(10 + 10\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 131 + 32\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0131321164
117618236312648
227601203042806104150816
3506651510105110410711015511134432
41067421113427056414611540812015065216128670784
5251686881375564687154200221214816351187235283012871804149248
66524202561112225497285800220588108243665683445670293081648748368124715835136
71839917254513582021586092235024876365351795028454847066262651503266417684786364591824970675712
8560593056605112176286808960889231778384526371267143203464955045102204393755272631041012823920120495560129680940570624
91834281043161614391543092311113669519395377551308743332513652141059432193017347543197827450965106217746875302383936517715567111143630343333888
1064092522397888811674470738892459215740894432442466658922337572092013095523596588951317011868923527149167672284220166490192396001369541799120251746792146932704319655936
112379488485269236116728659960560546337023891360662686383382811472233559867996145306294216597722748204294076547037459379562529945447343658428965723782068994156983004634769892312041899451523121152
1293460972943567060512841591748702944366432595363630811061991177703742947415506644904928994454876751715352441865413352580272025951648032750702730865232792084342293531420169502752159962813011120122644687411064193808892693297577984
133869662451082182032011257987637138196262815157198341888441888530595737786949340970925530368937510521093256552023060103954979403502783301728268060528717079853359683510086485500984035224245182137405320807807620329606014165173555655618905121514535352923617766637568
14168365050675723628633615824437500269871742841678710809707357472984920529720510723226542673601904668900505691624022037787660224834255124484291011602283770808082909558653858274704012252230455155104703305605670448224058359851015408868112324687685242226262026099136080252013761230048651493823155798016
Technical details
Node id
n-806f1ce683f83758 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-9f03a9f476d759af

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:             131,              32,               1;
  1:            1761,             823,              63,               1;
  2:           27601,           20304,            2806,             104,              1;
  3:          506651,          510105,          110410,            7110,            155,             1;
  4:        10674211,        13427056,         4146115,          408120,          15065,           216,           1;
  5:       251686881,       375564687,       154200221,        21481635,        1187235,         28301,         287,          1;
  6:      6524202561,     11222549728,      5800220588,      1082436656,       83445670,       2930816,       48748,        368,        1;
  7:    183991725451,    358202158609,    223502487636,     53517950284,     5484706626,     265150326,     6417684,      78636,      459,      1;
  8:   5605930566051,  12176286808960,   8892317783845,   2637126714320,   346495504510,   22043937552,   726310410,   12823920,   120495,    560,   1;
  9: 183428104316161, 439154309231111, 366951939537755, 130874333251365, 21410594321930, 1734754319782, 74509651062, 1774687530, 23839365, 177155, 671, 1;
%F Formula
The row polynomials satisfy P(n,x) = (11+x)*P(n-1,x) + (10+10*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 131 + 32*x + x^2.
%t Mathematica
P[0, x_] := 131 + 32*x + x^2;
P[n_, x_] := P[n, x] = Expand[(11 + x)*P[n - 1, x] + (10 + 10*x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 131 + 32*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + x)*v[i - 1] + (10 + 10*x)*deriv(v[i - 1], 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

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.