Real-rooted atlas

A109692 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(4 + 24\,x - 81\,x^{2} - 16\,n - 88\,n x + 270\,n x^{2} + 20\,n^{2} + 96\,n^{2} x - 252\,n^{2} x^{2} - 8\,n^{3} - 32\,n^{3} x + 72\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-24\,x^{2} - 36\,x^{3} + 88\,n x^{2} + 120\,n x^{3} - 96\,n^{2} x^{2} - 112\,n^{2} x^{3} + 32\,n^{3} x^{2} + 32\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(2 + 12\,x - 12\,n - 32\,n x + 8\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(-12\,x^{2} + 32\,n x^{2} - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
28402775
3484006122401300
43844544111969536262528285
538405926420397627449616035034020735946
646080880128391449672828166642100294818450935522223159
76451201471795280546688192427776240921600165614688592974488648640762819912
81032192027419443217854778885226756096836477160078537415684330059048130012416016418902529309635737
91857945605637390336426265390081480427192322894015976003461369569922596679163681195472194563092838633034459425001245620462382
10371589120012686819328010935292170244400684146688101830501792001484076685017614144844631792883494819123234933047694567943995170007922221807557995333805123
11817496064003103235112960300521649070081376401419960323684245525056006344323081228807353851677002885823417415362563116690623231201080439500636002192858052462019799007228002935082555121564
121961990553600819986064998408818443748638724532163039657984137886318631168002742859657393459237510499892987072360677725490094082443166162828683211442917065022400353314906286086864774454873184053443570205025160402384765730133
1351011754393600232816282435584027540375362076672157029322358390784535542047177676800120953425799450419219086071861918270722162594735841169408177643006814395843210515794741932624004380101168513101681219598427818966402039453781416415015495473018025009413150686590788658
141428329123020800706978285564723209125664017177640965718759043414622208216198518579695616005468883401811419136097891385194669187072127860269292113502208123639958896853296960886585608822099616004661938469373056731217489118030272394624443564957694794598068201391683246220048028060502296875590336506023027247215
Technical details
Node id
n-afb4312d6a5f8039 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b89c24c3b39f6ccd

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:         1;
  1:         2,          4;
  2:         8,         40,          27;
  3:        48,        400,         612,          240;
  4:       384,       4544,       11196,         9536,         2625;
  5:      3840,      59264,      203976,       274496,       160350,        34020;
  6:     46080,     880128,     3914496,      7282816,      6642100,      2948184,       509355;
  7:    645120,   14717952,    80546688,    192427776,    240921600,    165614688,     59297448,      8648640;
  8:  10321920,  274194432,  1785477888,   5226756096,   8364771600,   7853741568,   4330059048,   1300124160,   164189025;
  9: 185794560, 5637390336, 42626539008, 148042719232, 289401597600, 346136956992, 259667916368, 119547219456, 30928386330, 3445942500;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = -(8*n^3-20*n^2+16*n-4+(32*n^3-96*n^2+88*n-24)*x-(72*n^3-252*n^2+270*n-81)*x^2)*P(n-2,x) + ((32*n^3-96*n^2+88*n-24)*x^2+(32*n^3-112*n^2+120*n-36)*x^3)*d/dx P(n-2,x) + (8*n^2-12*n+2+(16*n^2-32*n+12)*x)*P(n-1,x) - (16*n^2-32*n+12)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 20*n^2 + 16*n - 4 + (32*n^3 - 96*n^2 + 88*n - 24)*x - (72*n^3 - 252*n^2 + 270*n - 81)*x^2)*P[n - 2, x] + ((32*n^3 - 96*n^2 + 88*n - 24)*x^2 + (32*n^3 - 112*n^2 + 120*n - 36)*x^3)*D[P[n - 2, x], x] + (8*n^2 - 12*n + 2 + (16*n^2 - 32*n + 12)*x)*P[n - 1, x] - (16*n^2 - 32*n + 12)*x^2*D[P[n - 1, x], x])/(2*n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 20*n^2 + 16*n - 4 + (32*n^3 - 96*n^2 + 88*n - 24)*x - (72*n^3 - 252*n^2 + 270*n - 81)*x^2)*v[i - 2] + ((32*n^3 - 96*n^2 + 88*n - 24)*x^2 + (32*n^3 - 112*n^2 + 120*n - 36)*x^3)*deriv(v[i - 2], x) + (8*n^2 - 12*n + 2 + (16*n^2 - 32*n + 12)*x)*v[i - 1] - (16*n^2 - 32*n + 12)*x^2*deriv(v[i - 1], x))/(2*n - 3)); 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.