Real-rooted atlas

A178821 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 6\,n x + 18\,n^{2} x + 9\,n^{3} x^{2}\right)\,P_{n}(x) = \left(32 + 144\,x + 216\,x^{2} + 108\,x^{3} + 8\,n + 180\,n x + 414\,n x^{2} + 243\,n x^{3} + 36\,n^{2} x + 162\,n^{2} x^{2} + 162\,n^{2} x^{3} + 18\,n^{3} x^{2} + 27\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1106070
26072012151995
32805040170101512037450
4112026880136080241920141750547750
540321209608164802177280255150011022486772500
6134404838404082400145152002551500022044960750141074156250
7422401774080179625607983360018711000024249456016503102046189440740437500

All 15 rows

Technical details
Node id
n-54a891875d98b29a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-56805ed74e931bf6

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:     10,       60;
  2:     60,      720,      1215;
  3:    280,     5040,     17010,      15120;
  4:   1120,    26880,    136080,     241920,     141750;
  5:   4032,   120960,    816480,    2177280,    2551500,     1102248;
  6:  13440,   483840,   4082400,   14515200,   25515000,    22044960,     7501410;
  7:  42240,  1774080,  17962560,   79833600,  187110000,   242494560,   165031020,    46189440;
  8: 126720,  6082560,  71850240,  383201280, 1122660000,  1939956480,  1980372240,  1108546560,  263063295;
  9: 366080, 19768320, 266872320, 1660538880, 5837832000, 12609717120, 17163226080, 14411105280, 6839645670, 1407334500;
%F Formula
The row polynomials satisfy (4*n+(18*n^2-6*n)*x+9*n^3*x^2)*P(n,x) = (8*n+32+(36*n^2+180*n+144)*x+(18*n^3+162*n^2+414*n+216)*x^2+(27*n^3+162*n^2+243*n+108)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 32 + (36*n^2 + 180*n + 144)*x + (18*n^3 + 162*n^2 + 414*n + 216)*x^2 + (27*n^3 + 162*n^2 + 243*n + 108)*x^3)*P[n - 1, x])/(4*n + (18*n^2 - 6*n)*x + 9*n^3*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n + 32 + (36*n^2 + 180*n + 144)*x + (18*n^3 + 162*n^2 + 414*n + 216)*x^2 + (27*n^3 + 162*n^2 + 243*n + 108)*x^3)*v[i - 1])/(4*n + (18*n^2 - 6*n)*x + 9*n^3*x^2)); 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. [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.