Real-rooted atlas

A204420 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27 + 18\,x - 3\,x^{2} + 108\,n - 60\,n x + 8\,n x^{2} - 144\,n^{2} + 56\,n^{2} x - 4\,n^{2} x^{2} + 80\,n^{3} - 16\,n^{3} x - 16\,n^{4}\right)\,P_{n-2}(x) + \left(4 - 2\,x - 12\,n + 4\,n x + 8\,n^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
2924942
3225690405601380
4110253696027090672052582320
58930253192210269325086940011812556707771680
6108056025405903960378554715144698400262473752245320727651065778560
7182614682257140722589071890874055310156446606787155375788918130463513051081080200198718720

All 15 rows

Technical details
Node id
n-ff1221cd172491ae · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b56b614d84ca9e33

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:                1,                2;
  2:                9,               24,                9;
  3:              225,              690,              405,               60;
  4:            11025,            36960,            27090,             6720,             525;
  5:           893025,          3192210,          2693250,           869400,          118125,            5670;
  6:        108056025,        405903960,        378554715,        144698400,        26247375,         2245320,          72765;
  7:      18261468225,      71407225890,      71890874055,      31015644660,      6787155375,       788918130,       46351305,      1081080;
  8:    4108830350625,   16614469872000,   17782109244900,    8416337529600,   2108653296750,    299675376000,    24235110900,   1037836800,    18243225;
  9: 1187451971330625, 4941282024929250, 5562698553512100, 2835382688938800, 788247975264750, 129622710917700, 12947508674100, 770788418400, 25120920825, 344594250;
%F Formula
The row polynomials satisfy P(n,x) = -(16*n^4-80*n^3+144*n^2-108*n+27+(16*n^3-56*n^2+60*n-18)*x+(4*n^2-8*n+3)*x^2)*P(n-2,x) + (8*n^2-12*n+4+(4*n-2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (16*n^3 - 56*n^2 + 60*n - 18)*x + (4*n^2 - 8*n + 3)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 4 + (4*n - 2)*x)*P[n - 1, 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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 80*n^3 + 144*n^2 - 108*n + 27 + (16*n^3 - 56*n^2 + 60*n - 18)*x + (4*n^2 - 8*n + 3)*x^2)*v[i - 2] + (8*n^2 - 12*n + 4 + (4*n - 2)*x)*v[i - 1]); 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.