Real-rooted atlas

A207607 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(50 + 100\,x + 54\,x^{2} - 20\,n - 76\,n x - 41\,n x^{2} + 8\,n x^{3} - 15\,n^{2} x^{2} - 12\,n^{2} x^{3} + 2\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n}(x) = \left(120 + 306\,x + 240\,x^{2} + 56\,x^{3} + 4\,x^{4} - 30\,n - 109\,n x - 110\,n x^{2} - 34\,n x^{3} - 2\,n x^{4} - 5\,n^{2} x - 37\,n^{2} x^{2} - 29\,n^{2} x^{3} - 4\,n^{2} x^{4} + 3\,n^{3} x^{2} + 7\,n^{3} x^{3} + 2\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-25 - 170\,x - 333\,x^{2} - 240\,x^{3} - 56\,x^{4} - 4\,x^{5} - 20\,n x^{2} - 34\,n x^{3} - 16\,n x^{4} - 2\,n x^{5} + 5\,n^{2} x^{2} + 16\,n^{2} x^{3} + 13\,n^{2} x^{4} + 2\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
2818632
32166458140
4552222288410599
5144710972548135122521
637721963753285610801981410474
798766321358113024667518782731643066
825841967446938543243522513446299636018175565
96765575581567442125321669058073024402448845920710603
1017711166530509631792264731155429030164395409926408570222858708
114636847745416219442843768301689020837469693183063686488188106932411440264
121213931358568507238299016321187422594424765174596198590453289997950117488282645574627
13317811384103615633054336243044498282540492752255706921156516837679588771501422961527697528180831325
14832040108001624758833411181535216510463016601517011877902461933248237817896720120138000520023219448113430714970718
Technical details
Node id
n-b8e8dd32d6dc1ef7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-ce7e5f6c844635b7

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,     4;
  2:    8,    18,      6;
  3:   21,    66,     45,      8;
  4:   55,   222,    228,     84,     10;
  5:  144,   710,    972,    548,    135,    12;
  6:  377,  2196,   3753,   2856,   1080,   198,    14;
  7:  987,  6632,  13581,  13024,   6675,  1878,   273,   16;
  8: 2584, 19674,  46938,  54324,  35225, 13446,  2996,  360,  18;
  9: 6765, 57558, 156744, 212532, 166905, 80730, 24402, 4488, 459, 20;
%F Formula
The row polynomials satisfy -(20*n-50+(76*n-100)*x-(2*n^3-15*n^2-41*n+54)*x^2-(4*n^3-12*n^2+8*n)*x^3)*P(n,x) = -(30*n-120+(5*n^2+109*n-306)*x-(3*n^3-37*n^2-110*n+240)*x^2-(7*n^3-29*n^2-34*n+56)*x^3-(2*n^3-4*n^2-2*n+4)*x^4)*P(n-1,x) - (25+170*x-(5*n^2-20*n-333)*x^2-(16*n^2-34*n-240)*x^3-(13*n^2-16*n-56)*x^4-(2*n^2-2*n-4)*x^5)*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[(-(30*n - 120 + (5*n^2 + 109*n - 306)*x - (3*n^3 - 37*n^2 - 110*n + 240)*x^2 - (7*n^3 - 29*n^2 - 34*n + 56)*x^3 - (2*n^3 - 4*n^2 - 2*n + 4)*x^4)*P[n - 1, x] - (25 + 170*x - (5*n^2 - 20*n - 333)*x^2 - (16*n^2 - 34*n - 240)*x^3 - (13*n^2 - 16*n - 56)*x^4 - (2*n^2 - 2*n - 4)*x^5)*D[P[n - 1, x], x])/(-20*n + 50 - (76*n - 100)*x + (2*n^3 - 15*n^2 - 41*n + 54)*x^2 + (4*n^3 - 12*n^2 + 8*n)*x^3)]];
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] = (-(30*n - 120 + (5*n^2 + 109*n - 306)*x - (3*n^3 - 37*n^2 - 110*n + 240)*x^2 - (7*n^3 - 29*n^2 - 34*n + 56)*x^3 - (2*n^3 - 4*n^2 - 2*n + 4)*x^4)*v[i - 1] - (25 + 170*x - (5*n^2 - 20*n - 333)*x^2 - (16*n^2 - 34*n - 240)*x^3 - (13*n^2 - 16*n - 56)*x^4 - (2*n^2 - 2*n - 4)*x^5)*deriv(v[i - 1], x))/(-20*n + 50 - (76*n - 100)*x + (2*n^3 - 15*n^2 - 41*n + 54)*x^2 + (4*n^3 - 12*n^2 + 8*n)*x^3)); 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