Real-rooted atlas

A114687 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 + 14\,x + 8\,x^{2} - 8\,x^{3} + 2\,n + 27\,n x + 40\,n x^{2} + 2\,n x^{3} + 10\,n^{2} x + 54\,n^{2} x^{2} + 41\,n^{2} x^{3} + 4\,n^{2} x^{4} + 18\,n^{3} x^{2} + 43\,n^{3} x^{3} + 10\,n^{3} x^{4} + 12\,n^{4} x^{3} + 8\,n^{4} x^{4} + 2\,n^{5} x^{4}\right)\,P_{n}(x) = \left(4 + 14\,x + 8\,x^{2} - 8\,x^{3} + 14\,n + 103\,n x + 220\,n x^{2} + 146\,n x^{3} + 8\,n x^{4} + 8\,n x^{5} + 84\,n^{2} x + 390\,n^{2} x^{2} + 557\,n^{2} x^{3} + 256\,n^{2} x^{4} + 36\,n^{2} x^{5} + 150\,n^{3} x^{2} + 443\,n^{3} x^{3} + 362\,n^{3} x^{4} + 80\,n^{3} x^{5} + 92\,n^{4} x^{3} + 152\,n^{4} x^{4} + 56\,n^{4} x^{5} + 14\,n^{5} x^{4} + 12\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-1 - 12\,x - 49\,x^{2} - 92\,x^{3} - 86\,x^{4} - 40\,x^{5} - 8\,x^{6} - 12\,n x - 91\,n x^{2} - 256\,n x^{3} - 329\,n x^{4} - 188\,n x^{5} - 36\,n x^{6} - 30\,n^{2} x^{2} - 164\,n^{2} x^{3} - 314\,n^{2} x^{4} - 244\,n^{2} x^{5} - 64\,n^{2} x^{6} - 24\,n^{3} x^{3} - 92\,n^{3} x^{4} - 108\,n^{3} x^{5} - 40\,n^{3} x^{6} - 4\,n^{4} x^{4} - 12\,n^{4} x^{5} - 8\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
13811
2115636103
345336432128941
41971936374423044008581
590311000284402688010000115278375
6427962184201420257920150000380163136716955
72079335100813668482213120176400070963213171281926565305

All 15 rows

Technical details
Node id
n-979aa417748eff6a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-dc15cf2bf87604ed

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:      3,        8;
  2:     11,       56,       36;
  3:     45,      336,      432,       128;
  4:    197,     1936,     3744,      2304,       400;
  5:    903,    11000,    28440,     26880,     10000,      1152;
  6:   4279,    62184,   201420,    257920,    150000,     38016,     3136;
  7:  20793,   351008,  1366848,   2213120,   1764000,    709632,   131712,     8192;
  8: 103049,  1981472,  9015552,  17669120,  17953600,   9999360,  2985472,   425984,   20736;
  9: 518859, 11194344, 58282416, 134163456, 166017600, 118734336, 49523712, 11501568, 1306368, 51200;
%F Formula
The row polynomials satisfy (2*n+4+(10*n^2+27*n+14)*x+(18*n^3+54*n^2+40*n+8)*x^2+(12*n^4+43*n^3+41*n^2+2*n-8)*x^3+(2*n^5+8*n^4+10*n^3+4*n^2)*x^4)*P(n,x) = (14*n+4+(84*n^2+103*n+14)*x+(150*n^3+390*n^2+220*n+8)*x^2+(92*n^4+443*n^3+557*n^2+146*n-8)*x^3+(14*n^5+152*n^4+362*n^3+256*n^2+8*n)*x^4+(12*n^5+56*n^4+80*n^3+36*n^2+8*n)*x^5)*P(n-1,x) - (1+(12*n+12)*x+(30*n^2+91*n+49)*x^2+(24*n^3+164*n^2+256*n+92)*x^3+(4*n^4+92*n^3+314*n^2+329*n+86)*x^4+(12*n^4+108*n^3+244*n^2+188*n+40)*x^5+(8*n^4+40*n^3+64*n^2+36*n+8)*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[((14*n + 4 + (84*n^2 + 103*n + 14)*x + (150*n^3 + 390*n^2 + 220*n + 8)*x^2 + (92*n^4 + 443*n^3 + 557*n^2 + 146*n - 8)*x^3 + (14*n^5 + 152*n^4 + 362*n^3 + 256*n^2 + 8*n)*x^4 + (12*n^5 + 56*n^4 + 80*n^3 + 36*n^2 + 8*n)*x^5)*P[n - 1, x] - (1 + (12*n + 12)*x + (30*n^2 + 91*n + 49)*x^2 + (24*n^3 + 164*n^2 + 256*n + 92)*x^3 + (4*n^4 + 92*n^3 + 314*n^2 + 329*n + 86)*x^4 + (12*n^4 + 108*n^3 + 244*n^2 + 188*n + 40)*x^5 + (8*n^4 + 40*n^3 + 64*n^2 + 36*n + 8)*x^6)*D[P[n - 1, x], x])/(2*n + 4 + (10*n^2 + 27*n + 14)*x + (18*n^3 + 54*n^2 + 40*n + 8)*x^2 + (12*n^4 + 43*n^3 + 41*n^2 + 2*n - 8)*x^3 + (2*n^5 + 8*n^4 + 10*n^3 + 4*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] = ((14*n + 4 + (84*n^2 + 103*n + 14)*x + (150*n^3 + 390*n^2 + 220*n + 8)*x^2 + (92*n^4 + 443*n^3 + 557*n^2 + 146*n - 8)*x^3 + (14*n^5 + 152*n^4 + 362*n^3 + 256*n^2 + 8*n)*x^4 + (12*n^5 + 56*n^4 + 80*n^3 + 36*n^2 + 8*n)*x^5)*v[i - 1] - (1 + (12*n + 12)*x + (30*n^2 + 91*n + 49)*x^2 + (24*n^3 + 164*n^2 + 256*n + 92)*x^3 + (4*n^4 + 92*n^3 + 314*n^2 + 329*n + 86)*x^4 + (12*n^4 + 108*n^3 + 244*n^2 + 188*n + 40)*x^5 + (8*n^4 + 40*n^3 + 64*n^2 + 36*n + 8)*x^6)*deriv(v[i - 1], x))/(2*n + 4 + (10*n^2 + 27*n + 14)*x + (18*n^3 + 54*n^2 + 40*n + 8)*x^2 + (12*n^4 + 43*n^3 + 41*n^2 + 2*n - 8)*x^3 + (2*n^5 + 8*n^4 + 10*n^3 + 4*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