Real-rooted atlas

A232535 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 + 6\,x + 9\,x^{2} - 20\,n - 2\,n x + 3\,n x^{2} + 32\,n^{2} - 88\,n^{2} x - 57\,n^{2} x^{2} - 16\,n^{3} + 172\,n^{3} x + 27\,n^{3} x^{2} + 9\,n^{3} x^{3} - 72\,n^{4} x + 120\,n^{4} x^{2} + 36\,n^{4} x^{3} - 36\,n^{5} x^{2} + 36\,n^{5} x^{3}\right)\,P_{n}(x) = \left(16 + 152\,x + 456\,x^{2} + 342\,x^{3} - 27\,x^{4} - 80\,n - 664\,n x - 1668\,n x^{2} - 1098\,n x^{3} - 135\,n x^{4} + 128\,n^{2} + 736\,n^{2} x + 966\,n^{2} x^{2} - 135\,n^{2} x^{4} - 64\,n^{3} + 112\,n^{3} x + 1572\,n^{3} x^{2} + 2007\,n^{3} x^{3} + 837\,n^{3} x^{4} - 288\,n^{4} x - 600\,n^{4} x^{2} + 756\,n^{4} x^{3} + 1080\,n^{4} x^{4} - 144\,n^{5} x^{2} - 180\,n^{5} x^{3} + 324\,n^{5} x^{4}\right)\,P_{n-1}(x) + \left(8\,x + 76\,x^{2} + 210\,x^{3} + 144\,x^{4} - 54\,x^{5} - 32\,n x - 256\,n x^{2} - 678\,n x^{3} - 720\,n x^{4} - 378\,n x^{5} + 32\,n^{2} x + 112\,n^{2} x^{2} - 186\,n^{2} x^{3} - 1044\,n^{2} x^{4} - 1026\,n^{2} x^{5} + 144\,n^{3} x^{2} + 480\,n^{3} x^{3} - 144\,n^{3} x^{4} - 864\,n^{3} x^{5} + 72\,n^{4} x^{3} + 144\,n^{4} x^{4} - 216\,n^{4} x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
132427
212168243423
348960299717285733
4192499223976328321012572117
57682457615940835251227337552488863127
630721167369538562930688379890019070642500479960363
7122885406725329152210124803868560033312384117522091119744111764529

All 15 rows

Technical details
Node id
n-c91b23b261d98e03 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b44bae31b520ea3b

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,       24;
  2:     12,      168,       243;
  3:     48,      960,      2997,      1728;
  4:    192,     4992,     23976,     32832,      10125;
  5:    768,    24576,    159408,    352512,     273375,      52488;
  6:   3072,   116736,    953856,   2930688,    3798900,    1907064,     250047;
  7:  12288,   540672,   5329152,  21012480,   38685600,   33312384,   11752209,    1119744;
  8:  49152,  2457600,  28366848, 136691712,  327888000,  408146688,  252047376,   66064896,   4782969;
  9: 196608, 11010048, 145649664, 829882368, 2459808000, 4040036352, 3658973472, 1706489856, 345968091, 19683000;
%F Formula
The row polynomials satisfy -(16*n^3-32*n^2+20*n-4+(72*n^4-172*n^3+88*n^2+2*n-6)*x+(36*n^5-120*n^4-27*n^3+57*n^2-3*n-9)*x^2-(36*n^5+36*n^4+9*n^3)*x^3)*P(n,x) = -(64*n^3-128*n^2+80*n-16+(288*n^4-112*n^3-736*n^2+664*n-152)*x+(144*n^5+600*n^4-1572*n^3-966*n^2+1668*n-456)*x^2+(180*n^5-756*n^4-2007*n^3+1098*n-342)*x^3-(324*n^5+1080*n^4+837*n^3-135*n^2-135*n-27)*x^4)*P(n-1,x) + ((32*n^2-32*n+8)*x+(144*n^3+112*n^2-256*n+76)*x^2+(72*n^4+480*n^3-186*n^2-678*n+210)*x^3+(144*n^4-144*n^3-1044*n^2-720*n+144)*x^4-(216*n^4+864*n^3+1026*n^2+378*n+54)*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[(-(64*n^3 - 128*n^2 + 80*n - 16 + (288*n^4 - 112*n^3 - 736*n^2 + 664*n - 152)*x + (144*n^5 + 600*n^4 - 1572*n^3 - 966*n^2 + 1668*n - 456)*x^2 + (180*n^5 - 756*n^4 - 2007*n^3 + 1098*n - 342)*x^3 - (324*n^5 + 1080*n^4 + 837*n^3 - 135*n^2 - 135*n - 27)*x^4)*P[n - 1, x] + ((32*n^2 - 32*n + 8)*x + (144*n^3 + 112*n^2 - 256*n + 76)*x^2 + (72*n^4 + 480*n^3 - 186*n^2 - 678*n + 210)*x^3 + (144*n^4 - 144*n^3 - 1044*n^2 - 720*n + 144)*x^4 - (216*n^4 + 864*n^3 + 1026*n^2 + 378*n + 54)*x^5)*D[P[n - 1, x], x])/(-16*n^3 + 32*n^2 - 20*n + 4 - (72*n^4 - 172*n^3 + 88*n^2 + 2*n - 6)*x - (36*n^5 - 120*n^4 - 27*n^3 + 57*n^2 - 3*n - 9)*x^2 + (36*n^5 + 36*n^4 + 9*n^3)*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] = (-(64*n^3 - 128*n^2 + 80*n - 16 + (288*n^4 - 112*n^3 - 736*n^2 + 664*n - 152)*x + (144*n^5 + 600*n^4 - 1572*n^3 - 966*n^2 + 1668*n - 456)*x^2 + (180*n^5 - 756*n^4 - 2007*n^3 + 1098*n - 342)*x^3 - (324*n^5 + 1080*n^4 + 837*n^3 - 135*n^2 - 135*n - 27)*x^4)*v[i - 1] + ((32*n^2 - 32*n + 8)*x + (144*n^3 + 112*n^2 - 256*n + 76)*x^2 + (72*n^4 + 480*n^3 - 186*n^2 - 678*n + 210)*x^3 + (144*n^4 - 144*n^3 - 1044*n^2 - 720*n + 144)*x^4 - (216*n^4 + 864*n^3 + 1026*n^2 + 378*n + 54)*x^5)*deriv(v[i - 1], x))/(-16*n^3 + 32*n^2 - 20*n + 4 - (72*n^4 - 172*n^3 + 88*n^2 + 2*n - 6)*x - (36*n^5 - 120*n^4 - 27*n^3 + 57*n^2 - 3*n - 9)*x^2 + (36*n^5 + 36*n^4 + 9*n^3)*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