Real-rooted atlas

A166979 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 - 4\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(-18\,x + 4\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(-81\,x^{2} + 36\,x^{3}\right)\,P_{n-2}''(x) + \left(-7 + 4\,x\right)\,P_{n-1}(x) + \left(18\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1 + 44\,x + 9\,x^{2}\)
\(P_{2}(x) = 1 + 444\,x + 270\,x^{2} + 16\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
1144954
2144427016731
31444461299282511527
414444412645036880237536210186
51444444250254912574401455005076494355059
61444444448548430396572967348250443016960464100451105
7144444444932420169119671260833384977530960216113121416640812539535148
81444444444178159832103516558888014222305125190491361210435706425424642697310069660161873
914444444444339503680989101630934768058117152225010810614553283324376032990208005186835415001212058171827755
101444444444446460569938790290602238345762309132967900058097696865126054285440283002051500875708715598065006122614465102553597384
11144444444444412285082883700982517172503708889978579786452530050092031731241206311994848271207677312093262076622173880850017437431872641692192767123678333
1214444444444444233516574790317023323209774526480345814053413578751511873388352126826753623012596482273934026158081024060375938625772066320036907645062947838412083519678276216940415201
13144444444444444443781492101602296572012794607915201315954512902939000745258982333533668167723203244073477180475285282406401035358849620810337758150730006477015725187340153184477671741632682252949677889878533158
1414444444444444448432848349930443501848053031238689185649717193976566229750361768876859227808881023915645993357317213741088667792520329842259931587369040478376379870009980957933746815048337187521382110730074664240216000256116934178346289121199
151444444444444444416023411864867842649518954021786820606368186841200574605501927517357295928787032916886123943690910747540021013165595848057766408923956284576270223453432068186383000139805552616738582692650200123523272590268402248395935201131435002805762894862226046867359360787
Technical details
Node id
n-bfbd37281ac07ff8 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-3b0795cfa52da853

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 1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1,          44,             9;
   2: 1,         444,           270,             16;
   3: 1,        4444,          6129,            928,             25;
   4: 1,       44444,        126450,          36880,           2375,            36;
   5: 1,      444444,       2502549,        1257440,         145500,          5076,           49;
   6: 1,     4444444,      48548430,       39657296,        7348250,        443016,         9604,          64;
   7: 1,    44444444,     932420169,     1196712608,      333849775,      30960216,      1131214,       16640,        81;
   8: 1,   444444444,   17815983210,    35165588880,    14222305125,    1904913612,    104357064,     2542464,     26973,     100;
   9: 1,  4444444444,  339503680989,  1016309347680,   581171522250,  108106145532,   8332437603,   299020800,   5186835,   41500,   121;
  10: 1, 44444444444, 6460569938790, 29060223834576, 23091329679000, 5809769686512, 605428544028, 30020515008, 757087155, 9806500, 61226, 144;
%F Formula
The row polynomials satisfy P(n,x) = (8-4*x+9*x^2)*P(n-2,x) - (18*x-4*x^2-4*x^3)*d/dx P(n-2,x) - (81*x^2-36*x^3)*d^2/dx^2 P(n-2,x) - (7-4*x)*P(n-1,x) + (18*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 44*x + 9*x^2, P(2,x) = 1 + 444*x + 270*x^2 + 16*x^3.
%t Mathematica
P[1, x_] := 1 + 44*x + 9*x^2;
P[2, x_] := 1 + 444*x + 270*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[(8 - 4*x + 9*x^2)*P[n - 2, x] - (18*x - 4*x^2 - 4*x^3)*D[P[n - 2, x], x] - (81*x^2 - 36*x^3)*D[P[n - 2, x], {x, 2}] - (7 - 4*x)*P[n - 1, x] + (18*x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 44*x + 9*x^2; v[2] = 1 + 444*x + 270*x^2 + 16*x^3; for(n = 3, N, my(i = n); v[i] = (8 - 4*x + 9*x^2)*v[i - 2] - (18*x - 4*x^2 - 4*x^3)*deriv(v[i - 2], x) - (81*x^2 - 36*x^3)*deriv(deriv(v[i - 2], x), x) - (7 - 4*x)*v[i - 1] + (18*x - 4*x^2)*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, v[i])))};
T(10)

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.