Real-rooted atlas

A060187 → γ → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(-96\,x + 192\,x^{2} - n - 16\,n x - 880\,n x^{2} + n^{2} + 64\,n^{2} x + 1328\,n^{2} x^{2} - 16\,n^{3} x - 784\,n^{3} x^{2} + 144\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(6\,x - 96\,x^{2} + 288\,x^{3} + 4\,n x - 32\,n x^{2} - 1216\,n x^{3} - 4\,n^{2} x + 32\,n^{2} x^{2} + 1632\,n^{2} x^{3} + 16\,n^{3} x^{2} - 768\,n^{3} x^{3} + 64\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(4\,x^{2} - 32\,x^{3} + 64\,x^{4} + 4\,n x^{2} + 48\,n x^{3} - 256\,n x^{4} - 4\,n^{2} x^{2} - 64\,n^{2} x^{3} + 320\,n^{2} x^{4} + 32\,n^{3} x^{3} - 128\,n^{3} x^{4}\right)\,P_{n-2}''(x) + \left(1 - 32\,x - n + 80\,n x - 64\,n^{2} x + 16\,n^{3} x\right)\,P_{n-1}(x) + \left(-6\,n x - 16\,n x^{2} + 4\,n^{2} x + 32\,n^{2} x^{2} - 16\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211
311617
418081
512887201009
6192887849713
7128646897662464134305
81868844798414182401874913
9126176263088019484672886400031005729
1017865614567904211226624323334400549207585
1112361127779484819965562886870508800186240614410807502193
1217084964059743041729514086411174848640099634728960229085039025
13121256642087259696141274484736154620952960030269958635525424557465605259025009809
1416377184106276082401107663278080192136789440006892162206105640012546330624129266144599185
15119131760537693824168433809873920221383378912000131149179625881616458829292953602090451400130563396290842867329
Technical details
Node id
n-a843ab95b98f04cf · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-98b8e10712046853

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1;
   3: 1,    16;
   4: 1,    80;
   5: 1,   288,      720;
   6: 1,   928,     8784;
   7: 1,  2864,    68976,     62464;
   8: 1,  8688,   447984,   1418240;
   9: 1, 26176,  2630880,  19484672,   8864000;
  10: 1, 78656, 14567904, 211226624, 323334400;
%F Formula
The row polynomials satisfy (n^2-2*n+1)*P(n,x) = (n^2-n-(16*n^3-64*n^2+16*n+96)*x+(144*n^4-784*n^3+1328*n^2-880*n+192)*x^2)*P(n-2,x) - ((4*n^2-4*n-6)*x-(16*n^3+32*n^2-32*n-96)*x^2-(64*n^4-768*n^3+1632*n^2-1216*n+288)*x^3)*d/dx P(n-2,x) - ((4*n^2-4*n-4)*x^2-(32*n^3-64*n^2+48*n-32)*x^3+(128*n^3-320*n^2+256*n-64)*x^4)*d^2/dx^2 P(n-2,x) - (n-1-(16*n^3-64*n^2+80*n-32)*x)*P(n-1,x) + ((4*n^2-6*n)*x-(16*n^3-32*n^2+16*n)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - n - (16*n^3 - 64*n^2 + 16*n + 96)*x + (144*n^4 - 784*n^3 + 1328*n^2 - 880*n + 192)*x^2)*P[n - 2, x] - ((4*n^2 - 4*n - 6)*x - (16*n^3 + 32*n^2 - 32*n - 96)*x^2 - (64*n^4 - 768*n^3 + 1632*n^2 - 1216*n + 288)*x^3)*D[P[n - 2, x], x] - ((4*n^2 - 4*n - 4)*x^2 - (32*n^3 - 64*n^2 + 48*n - 32)*x^3 + (128*n^3 - 320*n^2 + 256*n - 64)*x^4)*D[P[n - 2, x], {x, 2}] - (n - 1 - (16*n^3 - 64*n^2 + 80*n - 32)*x)*P[n - 1, x] + ((4*n^2 - 6*n)*x - (16*n^3 - 32*n^2 + 16*n)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((n^2 - n - (16*n^3 - 64*n^2 + 16*n + 96)*x + (144*n^4 - 784*n^3 + 1328*n^2 - 880*n + 192)*x^2)*v[i - 2] - ((4*n^2 - 4*n - 6)*x - (16*n^3 + 32*n^2 - 32*n - 96)*x^2 - (64*n^4 - 768*n^3 + 1632*n^2 - 1216*n + 288)*x^3)*deriv(v[i - 2], x) - ((4*n^2 - 4*n - 4)*x^2 - (32*n^3 - 64*n^2 + 48*n - 32)*x^3 + (128*n^3 - 320*n^2 + 256*n - 64)*x^4)*deriv(deriv(v[i - 2], x), x) - (n - 1 - (16*n^3 - 64*n^2 + 80*n - 32)*x)*v[i - 1] + ((4*n^2 - 6*n)*x - (16*n^3 - 32*n^2 + 16*n)*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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.