Real-rooted atlas

A229032 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(144\,n + 162\,n x + 25\,n x^{2} - 44\,n x^{3} + 240\,n^{2} x + 104\,n^{2} x^{2} - 26\,n^{2} x^{3} + 184\,n^{3} x^{2} + 78\,n^{3} x^{3} + 4\,n^{3} x^{4} + 48\,n^{4} x^{3} + 8\,n^{4} x^{4} + 4\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-432 - 1782\,x - 2547\,x^{2} - 1420\,x^{3} - 272\,x^{4} - 64\,x^{5} + 720\,n + 882\,n x - 1039\,n x^{2} - 1982\,n x^{3} - 888\,n x^{4} - 128\,n x^{5} + 1704\,n^{2} x + 3004\,n^{2} x^{2} + 1332\,n^{2} x^{3} + 36\,n^{2} x^{4} - 16\,n^{2} x^{5} + 1208\,n^{3} x^{2} + 1894\,n^{3} x^{3} + 788\,n^{3} x^{4} + 112\,n^{3} x^{5} + 288\,n^{4} x^{3} + 348\,n^{4} x^{4} + 80\,n^{4} x^{5} + 20\,n^{5} x^{4} + 16\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-216 - 1188\,x - 2610\,x^{2} - 2942\,x^{3} - 1848\,x^{4} - 672\,x^{5} - 128\,x^{6} - 864\,n x - 3720\,n x^{2} - 6292\,n x^{3} - 5276\,n x^{4} - 2224\,n x^{5} - 384\,n x^{6} - 888\,n^{2} x^{2} - 3116\,n^{2} x^{3} - 3932\,n^{2} x^{4} - 2120\,n^{2} x^{5} - 416\,n^{2} x^{6} - 288\,n^{3} x^{3} - 816\,n^{3} x^{4} - 720\,n^{3} x^{5} - 192\,n^{3} x^{6} - 24\,n^{4} x^{4} - 56\,n^{4} x^{5} - 32\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
122
271623
3206484
461288144493
518210888642134
654740164752102410339
716401420821888819245928

All 15 rows

Technical details
Node id
n-8aafd380e3a0603f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-f105a98b26320007

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 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;
  2:     7,     16;
  3:    20,     64;
  4:    61,    288,    144;
  5:   182,   1088,    864;
  6:   547,   4016,   4752,   1024;
  7:  1640,  14208,  21888,   8192;
  8:  4921,  49216,  94176,  53248,  6400;
  9: 14762, 167296, 381888, 286720, 64000;
%F Formula
The row polynomials satisfy (144*n+(240*n^2+162*n)*x+(184*n^3+104*n^2+25*n)*x^2+(48*n^4+78*n^3-26*n^2-44*n)*x^3+(4*n^5+8*n^4+4*n^3)*x^4)*P(n,x) = (720*n-432+(1704*n^2+882*n-1782)*x+(1208*n^3+3004*n^2-1039*n-2547)*x^2+(288*n^4+1894*n^3+1332*n^2-1982*n-1420)*x^3+(20*n^5+348*n^4+788*n^3+36*n^2-888*n-272)*x^4+(16*n^5+80*n^4+112*n^3-16*n^2-128*n-64)*x^5)*P(n-1,x) - (216+(864*n+1188)*x+(888*n^2+3720*n+2610)*x^2+(288*n^3+3116*n^2+6292*n+2942)*x^3+(24*n^4+816*n^3+3932*n^2+5276*n+1848)*x^4+(56*n^4+720*n^3+2120*n^2+2224*n+672)*x^5+(32*n^4+192*n^3+416*n^2+384*n+128)*x^6)*d/dx 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[((720*n - 432 + (1704*n^2 + 882*n - 1782)*x + (1208*n^3 + 3004*n^2 - 1039*n - 2547)*x^2 + (288*n^4 + 1894*n^3 + 1332*n^2 - 1982*n - 1420)*x^3 + (20*n^5 + 348*n^4 + 788*n^3 + 36*n^2 - 888*n - 272)*x^4 + (16*n^5 + 80*n^4 + 112*n^3 - 16*n^2 - 128*n - 64)*x^5)*P[n - 1, x] - (216 + (864*n + 1188)*x + (888*n^2 + 3720*n + 2610)*x^2 + (288*n^3 + 3116*n^2 + 6292*n + 2942)*x^3 + (24*n^4 + 816*n^3 + 3932*n^2 + 5276*n + 1848)*x^4 + (56*n^4 + 720*n^3 + 2120*n^2 + 2224*n + 672)*x^5 + (32*n^4 + 192*n^3 + 416*n^2 + 384*n + 128)*x^6)*D[P[n - 1, x], x])/(144*n + (240*n^2 + 162*n)*x + (184*n^3 + 104*n^2 + 25*n)*x^2 + (48*n^4 + 78*n^3 - 26*n^2 - 44*n)*x^3 + (4*n^5 + 8*n^4 + 4*n^3)*x^4)]];
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] = ((720*n - 432 + (1704*n^2 + 882*n - 1782)*x + (1208*n^3 + 3004*n^2 - 1039*n - 2547)*x^2 + (288*n^4 + 1894*n^3 + 1332*n^2 - 1982*n - 1420)*x^3 + (20*n^5 + 348*n^4 + 788*n^3 + 36*n^2 - 888*n - 272)*x^4 + (16*n^5 + 80*n^4 + 112*n^3 - 16*n^2 - 128*n - 64)*x^5)*v[i - 1] - (216 + (864*n + 1188)*x + (888*n^2 + 3720*n + 2610)*x^2 + (288*n^3 + 3116*n^2 + 6292*n + 2942)*x^3 + (24*n^4 + 816*n^3 + 3932*n^2 + 5276*n + 1848)*x^4 + (56*n^4 + 720*n^3 + 2120*n^2 + 2224*n + 672)*x^5 + (32*n^4 + 192*n^3 + 416*n^2 + 384*n + 128)*x^6)*deriv(v[i - 1], x))/(144*n + (240*n^2 + 162*n)*x + (184*n^3 + 104*n^2 + 25*n)*x^2 + (48*n^4 + 78*n^3 - 26*n^2 - 44*n)*x^3 + (4*n^5 + 8*n^4 + 4*n^3)*x^4)); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023