Real-rooted atlas

A109692 → basis A013609 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
2202444
32005443601104
42272995214304672033248
5296321813124117444104961512001184384
64400643479552109242241700377613103040399168048942336
77358976715970562886416646167592967360652804646983681210809602306201600

All 15 rows

Technical details
Node id
n-d5b94f37cd400214 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-5a0e1b6ae8b7867a

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          0;
  1:          2;
  2:         20,          24;
  3:        200,         544,          360;
  4:       2272,        9952,        14304,         6720;
  5:      29632,      181312,       411744,       410496,        151200;
  6:     440064,     3479552,     10924224,     17003776,      13103040,       3991680;
  7:    7358976,    71597056,    288641664,    616759296,     736065280,     464698368,     121080960;
  8:  137097216,  1587091456,   7840134144,  21413815296,   34905518080,   33933523968,   18201738240,   4151347200;
  9: 2818695168, 37890256896, 222064078848, 740868089856, 1538386475520, 2034948569088, 1673661072384, 781991792640, 158789030400;
%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) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
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] = 0; v[2] = 2; 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.