Real-rooted atlas

A145596 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-16 + 20\,x - 4\,x^{2} + 12\,n + n x + 3\,n x^{2} + 4\,n^{2} - 37\,n^{2} x + 9\,n^{2} x^{2} + 11\,n^{3} x - 6\,n^{3} x^{2} + 5\,n^{4} x - 6\,n^{4} x^{2} + 3\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(32 - 72\,x + 48\,x^{2} - 8\,x^{3} + 24\,n + 2\,n x - 42\,n x^{2} + 16\,n x^{3} + 4\,n^{2} + 59\,n^{2} x - 51\,n^{2} x^{2} + 26\,n^{3} x + 28\,n^{3} x^{2} - 12\,n^{3} x^{3} + 5\,n^{4} x + 30\,n^{4} x^{2} + 9\,n^{4} x^{3} + 6\,n^{5} x^{2} + 12\,n^{5} x^{3} + n^{6} x^{2} + 3\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(12\,x - 28\,x^{2} + 20\,x^{3} - 4\,x^{4} + 8\,n x - 4\,n x^{2} - 10\,n x^{3} + 6\,n x^{4} + 22\,n^{2} x^{2} - 23\,n^{2} x^{3} + n^{2} x^{4} + 10\,n^{3} x^{2} + 2\,n^{3} x^{3} - 12\,n^{3} x^{4} + 9\,n^{4} x^{3} - 9\,n^{4} x^{4} + 2\,n^{5} x^{3} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
282432
320160108288
44060010803202040
57016805670448075012650
61123920211683136014000151272072
7168806463504150528126000362882744387296

All 15 rows

Technical details
Node id
n-47ae55b1467a65a0 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7f770609ac5879ac

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:   8,    24;
  3:  20,   160,    108;
  4:  40,   600,   1080,     320;
  5:  70,  1680,   5670,    4480,     750;
  6: 112,  3920,  21168,   31360,   14000,    1512;
  7: 168,  8064,  63504,  150528,  126000,   36288,    2744;
  8: 240, 15120, 163296,  564480,  756000,  408240,   82320,   4608;
  9: 330, 26400, 374220, 1774080, 3465000, 2993760, 1131900, 168960, 7290;
%F Formula
The row polynomials satisfy (4*n^2+12*n-16+(5*n^4+11*n^3-37*n^2+n+20)*x+(n^6+3*n^5-6*n^4-6*n^3+9*n^2+3*n-4)*x^2)*P(n,x) = (4*n^2+24*n+32+(5*n^4+26*n^3+59*n^2+2*n-72)*x+(n^6+6*n^5+30*n^4+28*n^3-51*n^2-42*n+48)*x^2+(3*n^6+12*n^5+9*n^4-12*n^3+16*n-8)*x^3)*P(n-1,x) + ((8*n+12)*x+(10*n^3+22*n^2-4*n-28)*x^2+(2*n^5+9*n^4+2*n^3-23*n^2-10*n+20)*x^3-(2*n^5+9*n^4+12*n^3-n^2-6*n+4)*x^4)*d/dx 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[((4*n^2 + 24*n + 32 + (5*n^4 + 26*n^3 + 59*n^2 + 2*n - 72)*x + (n^6 + 6*n^5 + 30*n^4 + 28*n^3 - 51*n^2 - 42*n + 48)*x^2 + (3*n^6 + 12*n^5 + 9*n^4 - 12*n^3 + 16*n - 8)*x^3)*P[n - 1, x] + ((8*n + 12)*x + (10*n^3 + 22*n^2 - 4*n - 28)*x^2 + (2*n^5 + 9*n^4 + 2*n^3 - 23*n^2 - 10*n + 20)*x^3 - (2*n^5 + 9*n^4 + 12*n^3 - n^2 - 6*n + 4)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 12*n - 16 + (5*n^4 + 11*n^3 - 37*n^2 + n + 20)*x + (n^6 + 3*n^5 - 6*n^4 - 6*n^3 + 9*n^2 + 3*n - 4)*x^2)]];
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] = ((4*n^2 + 24*n + 32 + (5*n^4 + 26*n^3 + 59*n^2 + 2*n - 72)*x + (n^6 + 6*n^5 + 30*n^4 + 28*n^3 - 51*n^2 - 42*n + 48)*x^2 + (3*n^6 + 12*n^5 + 9*n^4 - 12*n^3 + 16*n - 8)*x^3)*v[i - 1] + ((8*n + 12)*x + (10*n^3 + 22*n^2 - 4*n - 28)*x^2 + (2*n^5 + 9*n^4 + 2*n^3 - 23*n^2 - 10*n + 20)*x^3 - (2*n^5 + 9*n^4 + 12*n^3 - n^2 - 6*n + 4)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 12*n - 16 + (5*n^4 + 11*n^3 - 37*n^2 + n + 20)*x + (n^6 + 3*n^5 - 6*n^4 - 6*n^3 + 9*n^2 + 3*n - 4)*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

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.