Real-rooted atlas

A094638 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(36\,n x^{2} + 8\,n^{2} x - 72\,n^{2} x^{2} - n^{3} - 8\,n^{3} x + 36\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(16\,n x^{3} - 8\,n^{2} x^{2} - 32\,n^{2} x^{3} + 8\,n^{3} x^{2} + 16\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-1 - 8\,n x + 2\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(8\,n x^{2} - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
265672134
32436810447682204
412026081184418688960042860
572020448129744322688349600138240961440
650401776961460304502668884976007008768225792024434016
7403201703808172798567655884817793960022738176015161932841287680693811200

All 15 rows

Technical details
Node id
n-875a8cb841d3dde9 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7a9f2033c9d45936

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:       2,         8;
  2:       6,        56,         72;
  3:      24,       368,       1044,         768;
  4:     120,      2608,      11844,       18688,        9600;
  5:     720,     20448,     129744,      322688,      349600,       138240;
  6:    5040,    177696,    1460304,     5026688,     8497600,      7008768,      2257920;
  7:   40320,   1703808,   17279856,    76558848,   177939600,    227381760,    151619328,     41287680;
  8:  362880,  17914752,  216855792,  1180545536,  3515427600,   6146164224,   6316443392,   3540123648,   836075520;
  9: 3628800, 205274880, 2894105376, 18744840704, 68357119200, 152581525632, 213745457408, 183902109696, 89009197056, 18579456000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(8*n^3-8*n^2)*x-(36*n^3-72*n^2+36*n)*x^2)*P(n-2,x) + ((8*n^3-8*n^2)*x^2+(16*n^3-32*n^2+16*n)*x^3)*d/dx P(n-2,x) + (2*n^2-1+(8*n^2-8*n)*x)*P(n-1,x) - (8*n^2-8*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (8*n^3 - 8*n^2)*x - (36*n^3 - 72*n^2 + 36*n)*x^2)*P[n - 2, x] + ((8*n^3 - 8*n^2)*x^2 + (16*n^3 - 32*n^2 + 16*n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 1 + (8*n^2 - 8*n)*x)*P[n - 1, x] - (8*n^2 - 8*n)*x^2*D[P[n - 1, x], x])/(n - 1)]];
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] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (8*n^3 - 8*n^2)*x - (36*n^3 - 72*n^2 + 36*n)*x^2)*v[i - 2] + ((8*n^3 - 8*n^2)*x^2 + (16*n^3 - 32*n^2 + 16*n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 1 + (8*n^2 - 8*n)*x)*v[i - 1] - (8*n^2 - 8*n)*x^2*deriv(v[i - 1], x))/(n - 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

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.