Real-rooted atlas

A116414 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n^{2} + 8\,n^{2} x - 8\,n^{2} x^{2} + 16\,n^{3} x + 8\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-8\,x - 28\,x^{2} - 32\,x^{3} + 2\,n + 8\,n x - 44\,n x^{2} - 96\,n x^{3} - 16\,n x^{4} + 4\,n^{2} + 56\,n^{2} x + 160\,n^{2} x^{2} + 48\,n^{2} x^{3} + 16\,n^{3} x + 104\,n^{3} x^{2} + 80\,n^{3} x^{3} + 16\,n^{3} x^{4}\right)\,P_{n-1}(x) + \left(-x - 12\,x^{2} - 36\,x^{3} - 16\,x^{4} - 2\,n x - 32\,n x^{2} - 140\,n x^{3} - 96\,n x^{4} - 16\,n x^{5} - 8\,n^{2} x^{2} - 68\,n^{2} x^{3} - 64\,n^{2} x^{4} - 16\,n^{2} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1189
21163956
3124126160311
41322617366051635
51404441984365521848308
614867541601227016416765141221
7156954752030850663846890826240200913
81641281123206507519574432716627558488569965804
91721656188161219054742641111068150566410629092952404591595
101802079272642095801003968305098558018566576471398584097430321632427
1118825503792033762019252807224168178521602839927527568360146146663188640101150728
1219630695104051682534231681532857246861056969659101320304001118017235261270410363093469957657
131104363666880759275573328829885352109233408279976014496044400588604808441228672186535843334807762171552457
141112425185696107833091481285447003123215948871240185815646532802415658498253437657617021388286528753281076167959986667200
Technical details
Node id
n-3ce4427c6db47f61 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-cf6f3ad2b5156eab

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: 1,  8;
  2: 1, 16,   39;
  3: 1, 24,  126,   160;
  4: 1, 32,  261,   736,    605;
  5: 1, 40,  444,  1984,   3655,   2184;
  6: 1, 48,  675,  4160,  12270,  16416,    7651;
  7: 1, 56,  954,  7520,  30850,  66384,   68908,   26240;
  8: 1, 64, 1281, 12320,  65075, 195744,  327166,  275584,   88569;
  9: 1, 72, 1656, 18816, 121905, 474264, 1111068, 1505664, 1062909, 295240;
%F Formula
The row polynomials satisfy (4*n^2+2*n+(16*n^3+8*n^2)*x+(8*n^3-8*n^2)*x^2)*P(n,x) = (4*n^2+2*n+(16*n^3+56*n^2+8*n-8)*x+(104*n^3+160*n^2-44*n-28)*x^2+(80*n^3+48*n^2-96*n-32)*x^3+(16*n^3-16*n)*x^4)*P(n-1,x) - ((2*n+1)*x+(8*n^2+32*n+12)*x^2+(68*n^2+140*n+36)*x^3+(64*n^2+96*n+16)*x^4+(16*n^2+16*n)*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[((4*n^2 + 2*n + (16*n^3 + 56*n^2 + 8*n - 8)*x + (104*n^3 + 160*n^2 - 44*n - 28)*x^2 + (80*n^3 + 48*n^2 - 96*n - 32)*x^3 + (16*n^3 - 16*n)*x^4)*P[n - 1, x] - ((2*n + 1)*x + (8*n^2 + 32*n + 12)*x^2 + (68*n^2 + 140*n + 36)*x^3 + (64*n^2 + 96*n + 16)*x^4 + (16*n^2 + 16*n)*x^5)*D[P[n - 1, x], x])/(4*n^2 + 2*n + (16*n^3 + 8*n^2)*x + (8*n^3 - 8*n^2)*x^2)]];
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] = ((4*n^2 + 2*n + (16*n^3 + 56*n^2 + 8*n - 8)*x + (104*n^3 + 160*n^2 - 44*n - 28)*x^2 + (80*n^3 + 48*n^2 - 96*n - 32)*x^3 + (16*n^3 - 16*n)*x^4)*v[i - 1] - ((2*n + 1)*x + (8*n^2 + 32*n + 12)*x^2 + (68*n^2 + 140*n + 36)*x^3 + (64*n^2 + 96*n + 16)*x^4 + (16*n^2 + 16*n)*x^5)*deriv(v[i - 1], x))/(4*n^2 + 2*n + (16*n^3 + 8*n^2)*x + (8*n^3 - 8*n^2)*x^2)); 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