Real-rooted atlas

A085880 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n - 2\,n x + 2\,n^{2} x\right)\,P_{n}(x) = \left(-12 - 24\,x + 8\,n - 8\,n x - 48\,n x^{2} + 16\,n^{2} x + 32\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
22810
386496168
440480144012803240
5224358416128286721792066528
61344268801612804300805376002580481415232
7844820275215206405406720101376009732096378470430792960
8549121537536138378246150144015375360022140518417220403256229888680524416

All 15 rows

Technical details
Node id
n-9e0bde9ee5a1fd97 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7cabb77b45806852

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 -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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       2,        8;
   3:       8,       64,         96;
   4:      40,      480,       1440,       1280;
   5:     224,     3584,      16128,      28672,       17920;
   6:    1344,    26880,     161280,     430080,      537600,      258048;
   7:    8448,   202752,    1520640,    5406720,    10137600,     9732096,     3784704;
   8:   54912,  1537536,   13837824,   61501440,   153753600,   221405184,   172204032,    56229888;
   9:  366080, 11714560,  123002880,  656015360,  2050048000,  3936092160,  4592107520,  2998927360,   843448320;
  10: 2489344, 89616384, 1075396608, 6691356672, 25092587520, 60222210048, 93678993408, 91767177216, 51619037184, 12745441280;
%F Formula
The row polynomials satisfy (n+(2*n^2-2*n)*x)*P(n,x) = (8*n-12+(16*n^2-8*n-24)*x+(32*n^2-48*n)*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n - 12 + (16*n^2 - 8*n - 24)*x + (32*n^2 - 48*n)*x^2)*P[n - 1, x])/(n + (2*n^2 - 2*n)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n - 12 + (16*n^2 - 8*n - 24)*x + (32*n^2 - 48*n)*x^2)*v[i - 1])/(n + (2*n^2 - 2*n)*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

Same fingerprint ?

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.