Real-rooted atlas

A109692 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
228441284
32487446321201744
42656122241928011488168047328
533472210944502400547744253824302401578624
64861443919616126640002064518417353984661228866528062346496
7800409678956032324440192712973184890255104611911424198530688172972802842368000

All 15 rows

Technical details
Node id
n-ce9a944de361e9e5 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e2f5e9f7ec908abd

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,           2;
  2:         28,          44,           12;
  3:        248,         744,          632,          120;
  4:       2656,       12224,        19280,        11488,          1680;
  5:      33472,      210944,       502400,       547744,        253824,         30240;
  6:     486144,     3919616,     12664000,     20645184,      17353984,       6612288,        665280;
  7:    8004096,    78956032,    324440192,    712973184,     890255104,     611911424,     198530688,      17297280;
  8:  147419136,  1724188672,   8633679872,  24027193344,   40258971904,   40914627584,   23857325568,    6751595520,    518918400;
  9: 3004489728, 40708952064, 241009207296, 814889449472, 1723603497984, 2342625864192, 2012819167232, 1021086231552, 256538004480, 17643225600;
%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+(32*n^3-112*n^2+120*n-36)*x^2)*P(n-2,x) + (8*n^2-12*n+2+(16*n^2-32*n+12)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 2*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 + (32*n^3 - 112*n^2 + 120*n - 36)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 2 + (16*n^2 - 32*n + 12)*x)*P[n - 1, 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] = 4 + 2*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 + (32*n^3 - 112*n^2 + 120*n - 36)*x^2)*v[i - 2] + (8*n^2 - 12*n + 2 + (16*n^2 - 32*n + 12)*x)*v[i - 1])/(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.