Real-rooted atlas

A225471 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 + 16\,x + 9\,x^{2} + 32\,n - 16\,n x - 16\,n^{2}\right)\,P_{n-2}(x) + \left(-16\,x^{2} + 4\,x^{3} + 16\,n x^{2}\right)\,P_{n-2}'(x) + \left(-4 + 4\,x + 8\,n\right)\,P_{n-1}(x) + \left(-4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1448
23248989
3384704216161320
461441280050406402524649
512288028057612960021760150036556352
62949120722534437416967526407000030244914741873
782575360214106112121024512277258243136000185472548864448758832
826424115207181697024435452313611023810561436736001045094442806492168115435574641
9951268147202691107389441729216512004742709248068947200005831239682963520089088014580100592094682072
10380507258880011144936816640752236521062422044999680003498936320003325335552019791045127434240017107202200012125062076751337
1116742319390720050559751028736035606017710489611037109229977618839851008000196699447296013234211020858560184321693612803080000319441441160397189682200
12803631330754560024938373269422080182284828991815685930808300797952107676768010240012154512015360090297193226244539438858241554086476835692800052601124492816958341790143509625
1341788829199237120013289406632401305601003992450613641216340808223462260736652588073353216007870891707334656634981818318848353990011781121382647698432377465088007054087048596224615161963165431130278830800
1423401744351572787200760922303094167961605921369372665420185620870135981644185600418700605993779200053474261817360384046272139983060992281170807357440012223013197824038207769600008517616307213208832001353352082320225184348763032875602625
Technical details
Node id
n-424171701c31281c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e916b8d181fb7bc4

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,            4;
  2:          32,           48,            9;
  3:         384,          704,          216,          16;
  4:        6144,        12800,         5040,         640,         25;
  5:      122880,       280576,       129600,       21760,       1500,        36;
  6:     2949120,      7225344,      3741696,      752640,      70000,      3024,       49;
  7:    82575360,    214106112,    121024512,    27725824,    3136000,    185472,     5488,     64;
  8:  2642411520,   7181697024,   4354523136,  1102381056,  143673600,  10450944,   428064,   9216,    81;
  9: 95126814720, 269110738944, 172921651200, 47427092480, 6894720000, 583123968, 29635200, 890880, 14580, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^2-32*n+16+(16*n-16)*x-9*x^2)*P(n-2,x) + ((16*n-16)*x^2+4*x^3)*d/dx P(n-2,x) + (8*n-4+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^2 - 32*n + 16 + (16*n - 16)*x - 9*x^2)*P[n - 2, x] + ((16*n - 16)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (8*n - 4 + 4*x)*P[n - 1, x] - 4*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 32*n + 16 + (16*n - 16)*x - 9*x^2)*v[i - 2] + ((16*n - 16)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (8*n - 4 + 4*x)*v[i - 1] - 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 - 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.