Real-rooted atlas

A055151 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(72 + 96\,x + 24\,x^{2} - 24\,x^{3} + 36\,n + 168\,n x + 72\,n x^{2} - 26\,n x^{3} + 60\,n^{2} x + 122\,n^{2} x^{2} + 35\,n^{2} x^{3} + 2\,n^{2} x^{4} + 46\,n^{3} x^{2} + 45\,n^{3} x^{3} + 5\,n^{3} x^{4} + 12\,n^{4} x^{3} + 4\,n^{4} x^{4} + n^{5} x^{4}\right)\,P_{n}(x) = \left(-72 - 420\,x - 744\,x^{2} - 456\,x^{3} - 64\,x^{4} - 16\,x^{5} + 180\,n + 318\,n x - 120\,n x^{2} - 478\,n x^{3} - 240\,n x^{4} - 32\,n x^{5} + 426\,n^{2} x + 826\,n^{2} x^{2} + 411\,n^{2} x^{3} + 14\,n^{2} x^{4} - 4\,n^{2} x^{5} + 302\,n^{3} x^{2} + 493\,n^{3} x^{3} + 209\,n^{3} x^{4} + 28\,n^{3} x^{5} + 72\,n^{4} x^{3} + 88\,n^{4} x^{4} + 20\,n^{4} x^{5} + 5\,n^{5} x^{4} + 4\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-54 - 324\,x - 750\,x^{2} - 852\,x^{3} - 508\,x^{4} - 168\,x^{5} - 32\,x^{6} - 216\,n x - 954\,n x^{2} - 1644\,n x^{3} - 1386\,n x^{4} - 576\,n x^{5} - 96\,n x^{6} - 222\,n^{2} x^{2} - 788\,n^{2} x^{3} - 1004\,n^{2} x^{4} - 542\,n^{2} x^{5} - 104\,n^{2} x^{6} - 72\,n^{3} x^{3} - 204\,n^{3} x^{4} - 180\,n^{3} x^{5} - 48\,n^{3} x^{6} - 6\,n^{4} x^{4} - 14\,n^{4} x^{5} - 8\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2246
341216
49401867
52112090231
65136040580896
7127106415755603326

All 15 rows

Technical details
Node id
n-6fff870f5716a66e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-dab27b0021826ead

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:   1;
  2:   2,    4;
  3:   4,   12;
  4:   9,   40,    18;
  5:  21,  120,    90;
  6:  51,  360,   405,    80;
  7: 127, 1064,  1575,   560;
  8: 323, 3136,  5796,  3136,  350;
  9: 835, 9216, 20412, 14784, 3150;
%F Formula
The row polynomials satisfy (36*n+72+(60*n^2+168*n+96)*x+(46*n^3+122*n^2+72*n+24)*x^2+(12*n^4+45*n^3+35*n^2-26*n-24)*x^3+(n^5+4*n^4+5*n^3+2*n^2)*x^4)*P(n,x) = (180*n-72+(426*n^2+318*n-420)*x+(302*n^3+826*n^2-120*n-744)*x^2+(72*n^4+493*n^3+411*n^2-478*n-456)*x^3+(5*n^5+88*n^4+209*n^3+14*n^2-240*n-64)*x^4+(4*n^5+20*n^4+28*n^3-4*n^2-32*n-16)*x^5)*P(n-1,x) - (54+(216*n+324)*x+(222*n^2+954*n+750)*x^2+(72*n^3+788*n^2+1644*n+852)*x^3+(6*n^4+204*n^3+1004*n^2+1386*n+508)*x^4+(14*n^4+180*n^3+542*n^2+576*n+168)*x^5+(8*n^4+48*n^3+104*n^2+96*n+32)*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[((180*n - 72 + (426*n^2 + 318*n - 420)*x + (302*n^3 + 826*n^2 - 120*n - 744)*x^2 + (72*n^4 + 493*n^3 + 411*n^2 - 478*n - 456)*x^3 + (5*n^5 + 88*n^4 + 209*n^3 + 14*n^2 - 240*n - 64)*x^4 + (4*n^5 + 20*n^4 + 28*n^3 - 4*n^2 - 32*n - 16)*x^5)*P[n - 1, x] - (54 + (216*n + 324)*x + (222*n^2 + 954*n + 750)*x^2 + (72*n^3 + 788*n^2 + 1644*n + 852)*x^3 + (6*n^4 + 204*n^3 + 1004*n^2 + 1386*n + 508)*x^4 + (14*n^4 + 180*n^3 + 542*n^2 + 576*n + 168)*x^5 + (8*n^4 + 48*n^3 + 104*n^2 + 96*n + 32)*x^6)*D[P[n - 1, x], x])/(36*n + 72 + (60*n^2 + 168*n + 96)*x + (46*n^3 + 122*n^2 + 72*n + 24)*x^2 + (12*n^4 + 45*n^3 + 35*n^2 - 26*n - 24)*x^3 + (n^5 + 4*n^4 + 5*n^3 + 2*n^2)*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] = ((180*n - 72 + (426*n^2 + 318*n - 420)*x + (302*n^3 + 826*n^2 - 120*n - 744)*x^2 + (72*n^4 + 493*n^3 + 411*n^2 - 478*n - 456)*x^3 + (5*n^5 + 88*n^4 + 209*n^3 + 14*n^2 - 240*n - 64)*x^4 + (4*n^5 + 20*n^4 + 28*n^3 - 4*n^2 - 32*n - 16)*x^5)*v[i - 1] - (54 + (216*n + 324)*x + (222*n^2 + 954*n + 750)*x^2 + (72*n^3 + 788*n^2 + 1644*n + 852)*x^3 + (6*n^4 + 204*n^3 + 1004*n^2 + 1386*n + 508)*x^4 + (14*n^4 + 180*n^3 + 542*n^2 + 576*n + 168)*x^5 + (8*n^4 + 48*n^3 + 104*n^2 + 96*n + 32)*x^6)*deriv(v[i - 1], x))/(36*n + 72 + (60*n^2 + 168*n + 96)*x + (46*n^3 + 122*n^2 + 72*n + 24)*x^2 + (12*n^4 + 45*n^3 + 35*n^2 - 26*n - 24)*x^3 + (n^5 + 4*n^4 + 5*n^3 + 2*n^2)*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