Real-rooted atlas

A225471 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 + 48\,x + 81\,x^{2} + 32\,n - 48\,n x - 16\,n^{2}\right)\,P_{n-2}(x) + \left(-48\,x^{2} + 36\,x^{3} + 48\,n x^{2}\right)\,P_{n-2}'(x) + \left(-4 + 12\,x + 8\,n\right)\,P_{n-1}(x) + \left(-12\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141216
23214481257
3384211219444324872
461443840045360172802025109209
5122880841728116640058752012150087482848776
6294912021676032336752642032128056700007348323572185062249
78257536064231833610892206087485972482540160004506969640007521399682865937968
8264241152021545091072391907082242976428851211637561600253957939231205865620155392531441107652385809
995126814720807332216832155629486080012805314969605584723200001416991242242160406080019483545609565938019683004463106876576
1038050725888003343481044992067701286895616595214991360002834138419200080805653913601442767189248162586828800112240339204330260007144929202501636876593
11167423193907200151679253086208032045415939440642980019492093952152602793164800047797965692928096477398341632128071123107841111179358080606236400001886261256255091689983242624805496
12803631330754560074815119808266240164056346092634112160131824121544704872181820882944002953546419732480065826653861928969927752782970881019636137428487025413824000310604353488795886041689813529531477997960694121
13417888291992371200398682198972039168090359320555227709449201822033481039872528596339416104960019126266848823214084629017455544401927741761557653094490715515494123527429645327104004165367856249615227952929283269212455631248730830391231536451505880
1423401744351572787200228276690928250388480532923243539887816704563493671504393011200339147490854961152000129942456216185733120337323900476514631686149205556907212800801951895909232640752043529036800005029567253238528233990496230400719226740232013124466936010761680251857949320130960185945
Technical details
Node id
n-798a623cf2a55ac8 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-63357bf49833c535

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:           4,           12;
  2:          32,          144,            81;
  3:         384,         2112,          1944,           432;
  4:        6144,        38400,         45360,         17280,         2025;
  5:      122880,       841728,       1166400,        587520,       121500,         8748;
  6:     2949120,     21676032,      33675264,      20321280,      5670000,       734832,       35721;
  7:    82575360,    642318336,    1089220608,     748597248,    254016000,     45069696,     4000752,     139968;
  8:  2642411520,  21545091072,   39190708224,   29764288512,  11637561600,   2539579392,   312058656,   20155392,   531441;
  9: 95126814720, 807332216832, 1556294860800, 1280531496960, 558472320000, 141699124224, 21604060800, 1948354560, 95659380, 1968300;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^2-32*n+16+(48*n-48)*x-81*x^2)*P(n-2,x) + ((48*n-48)*x^2+36*x^3)*d/dx P(n-2,x) + (8*n-4+12*x)*P(n-1,x) - 12*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 12*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^2 - 32*n + 16 + (48*n - 48)*x - 81*x^2)*P[n - 2, x] + ((48*n - 48)*x^2 + 36*x^3)*D[P[n - 2, x], x] + (8*n - 4 + 12*x)*P[n - 1, x] - 12*x^2*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] = 1; v[2] = 4 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 32*n + 16 + (48*n - 48)*x - 81*x^2)*v[i - 2] + ((48*n - 48)*x^2 + 36*x^3)*deriv(v[i - 2], x) + (8*n - 4 + 12*x)*v[i - 1] - 12*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 - 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.