Real-rooted atlas

A081580 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(96 - 48\,x - 32\,x^{2} - 18\,x^{3} + 2\,x^{4} + 288\,n x - 24\,n x^{2} - 36\,n x^{3} - 4\,n x^{4} + 264\,n^{2} x^{2} + 60\,n^{2} x^{3} - 4\,n^{2} x^{4} - n^{2} x^{5} + 72\,n^{3} x^{3} + 12\,n^{3} x^{4} - n^{3} x^{5} + 6\,n^{4} x^{4}\right)\,P_{n}(x) = \left(480 + 1680\,x + 2240\,x^{2} + 1430\,x^{3} + 450\,x^{4} + 50\,x^{5} - 10\,x^{6} + 1440\,n x + 3960\,n x^{2} + 4020\,n x^{3} + 1740\,n x^{4} + 215\,n x^{5} - 25\,n x^{6} + 1320\,n^{2} x^{2} + 2700\,n^{2} x^{3} + 1720\,n^{2} x^{4} + 320\,n^{2} x^{5} - 20\,n^{2} x^{6} + 360\,n^{3} x^{3} + 540\,n^{3} x^{4} + 175\,n^{3} x^{5} - 5\,n^{3} x^{6} + 30\,n^{4} x^{4} + 30\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(192 + 288\,x + 80\,x^{2} - 140\,x^{3} - 60\,x^{4} + x^{5} - x^{6} + 576\,n x + 768\,n x^{2} + 384\,n x^{3} + 44\,n x^{4} - 3\,n x^{6} + 528\,n^{2} x^{2} + 600\,n^{2} x^{3} + 220\,n^{2} x^{4} + 15\,n^{2} x^{5} - 3\,n^{2} x^{6} + 144\,n^{3} x^{3} + 120\,n^{3} x^{4} + 22\,n^{3} x^{5} - n^{3} x^{6} + 12\,n^{4} x^{4} + 6\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
2936954
32816814416356
41018081116368252418
53423420666040807503615288
611891426835685317601087513324995158
740885723217524820656010700023904215664576252

All 15 rows

Technical details
Node id
n-d73a17b0e1a00142 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-df7b071a744cc9e5

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,      4;
  2:     9,     36,       9;
  3:    28,    168,     144,      16;
  4:   101,    808,    1116,     368,      25;
  5:   342,   3420,    6660,    4080,     750,      36;
  6:  1189,  14268,   35685,   31760,   10875,    1332,     49;
  7:  4088,  57232,  175248,  206560,  107000,   23904,   2156,    64;
  8: 14121, 225936,  818208, 1200672,  839250,  286848,  46158,  3264,   81;
  9: 48682, 876276, 3664872, 6446496, 5703300, 2672136, 656208, 81216, 4698, 100;
%F Formula
The row polynomials satisfy (96+(288*n-48)*x+(264*n^2-24*n-32)*x^2+(72*n^3+60*n^2-36*n-18)*x^3+(6*n^4+12*n^3-4*n^2-4*n+2)*x^4-(n^3+n^2)*x^5)*P(n,x) = (480+(1440*n+1680)*x+(1320*n^2+3960*n+2240)*x^2+(360*n^3+2700*n^2+4020*n+1430)*x^3+(30*n^4+540*n^3+1720*n^2+1740*n+450)*x^4+(30*n^4+175*n^3+320*n^2+215*n+50)*x^5-(5*n^3+20*n^2+25*n+10)*x^6)*P(n-2,x) + (192+(576*n+288)*x+(528*n^2+768*n+80)*x^2+(144*n^3+600*n^2+384*n-140)*x^3+(12*n^4+120*n^3+220*n^2+44*n-60)*x^4+(6*n^4+22*n^3+15*n^2+1)*x^5-(n^3+3*n^2+3*n+1)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((480 + (1440*n + 1680)*x + (1320*n^2 + 3960*n + 2240)*x^2 + (360*n^3 + 2700*n^2 + 4020*n + 1430)*x^3 + (30*n^4 + 540*n^3 + 1720*n^2 + 1740*n + 450)*x^4 + (30*n^4 + 175*n^3 + 320*n^2 + 215*n + 50)*x^5 - (5*n^3 + 20*n^2 + 25*n + 10)*x^6)*P[n - 2, x] + (192 + (576*n + 288)*x + (528*n^2 + 768*n + 80)*x^2 + (144*n^3 + 600*n^2 + 384*n - 140)*x^3 + (12*n^4 + 120*n^3 + 220*n^2 + 44*n - 60)*x^4 + (6*n^4 + 22*n^3 + 15*n^2 + 1)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*P[n - 1, x])/(96 + (288*n - 48)*x + (264*n^2 - 24*n - 32)*x^2 + (72*n^3 + 60*n^2 - 36*n - 18)*x^3 + (6*n^4 + 12*n^3 - 4*n^2 - 4*n + 2)*x^4 - (n^3 + n^2)*x^5)]];
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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((480 + (1440*n + 1680)*x + (1320*n^2 + 3960*n + 2240)*x^2 + (360*n^3 + 2700*n^2 + 4020*n + 1430)*x^3 + (30*n^4 + 540*n^3 + 1720*n^2 + 1740*n + 450)*x^4 + (30*n^4 + 175*n^3 + 320*n^2 + 215*n + 50)*x^5 - (5*n^3 + 20*n^2 + 25*n + 10)*x^6)*v[i - 2] + (192 + (576*n + 288)*x + (528*n^2 + 768*n + 80)*x^2 + (144*n^3 + 600*n^2 + 384*n - 140)*x^3 + (12*n^4 + 120*n^3 + 220*n^2 + 44*n - 60)*x^4 + (6*n^4 + 22*n^3 + 15*n^2 + 1)*x^5 - (n^3 + 3*n^2 + 3*n + 1)*x^6)*v[i - 1])/(96 + (288*n - 48)*x + (264*n^2 - 24*n - 32)*x^2 + (72*n^3 + 60*n^2 - 36*n - 18)*x^3 + (6*n^4 + 12*n^3 - 4*n^2 - 4*n + 2)*x^4 - (n^3 + n^2)*x^5)); 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.