Real-rooted atlas

A104684 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36 - 176\,x + 64\,x^{2} - 84\,n + 456\,n x - 192\,n x^{2} + 73\,n^{2} - 468\,n^{2} x + 240\,n^{2} x^{2} - 28\,n^{3} + 238\,n^{3} x - 160\,n^{3} x^{2} + 4\,n^{4} - 60\,n^{4} x + 60\,n^{4} x^{2} + 6\,n^{5} x - 12\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(108 - 132\,x - 214\,x^{2} + 20\,x^{3} - 270\,n + 496\,n x + 630\,n x^{2} - 100\,n x^{3} + 252\,n^{2} - 728\,n^{2} x - 701\,n^{2} x^{2} + 197\,n^{2} x^{3} - 104\,n^{3} + 520\,n^{3} x + 338\,n^{3} x^{2} - 196\,n^{3} x^{3} + 16\,n^{4} - 180\,n^{4} x - 43\,n^{4} x^{2} + 104\,n^{4} x^{3} + 24\,n^{5} x - 16\,n^{5} x^{2} - 28\,n^{5} x^{3} + 4\,n^{6} x^{2} + 3\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(27\,x - 32\,x^{2} - 55\,x^{3} + 4\,x^{4} - 54\,n x + 104\,n x^{2} + 138\,n x^{3} - 20\,n x^{4} + 36\,n^{2} x - 124\,n^{2} x^{2} - 123\,n^{2} x^{3} + 37\,n^{2} x^{4} - 8\,n^{3} x + 64\,n^{3} x^{2} + 40\,n^{3} x^{3} - 32\,n^{3} x^{4} - 12\,n^{4} x^{2} + n^{4} x^{3} + 13\,n^{4} x^{4} - 2\,n^{5} x^{3} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
36814
4309627153
5140720540641464
663044805670192012512825
727722520045360268805250216105678
8120121330563118502688009450012096343832657

All 15 rows

Technical details
Node id
n-3874d1cec4d3bc95 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e73e7a37f4a9c691

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      0;
   2:      1;
   3:      6,       8;
   4:     30,      96,       27;
   5:    140,     720,      540,       64;
   6:    630,    4480,     5670,     1920,      125;
   7:   2772,   25200,    45360,    26880,     5250,     216;
   8:  12012,  133056,   311850,   268800,    94500,   12096,    343;
   9:  51480,  672672,  1945944,  2217600,  1155000,  272160,  24696,   512;
  10: 218790, 3294720, 11351340, 16144128, 11261250, 3991680, 679140, 46080, 729;
%F Formula
The row polynomials satisfy (4*n^4-28*n^3+73*n^2-84*n+36+(6*n^5-60*n^4+238*n^3-468*n^2+456*n-176)*x+(n^6-12*n^5+60*n^4-160*n^3+240*n^2-192*n+64)*x^2)*P(n,x) = (16*n^4-104*n^3+252*n^2-270*n+108+(24*n^5-180*n^4+520*n^3-728*n^2+496*n-132)*x+(4*n^6-16*n^5-43*n^4+338*n^3-701*n^2+630*n-214)*x^2+(3*n^6-28*n^5+104*n^4-196*n^3+197*n^2-100*n+20)*x^3)*P(n-1,x) - ((8*n^3-36*n^2+54*n-27)*x+(12*n^4-64*n^3+124*n^2-104*n+32)*x^2+(2*n^5-n^4-40*n^3+123*n^2-138*n+55)*x^3+(2*n^5-13*n^4+32*n^3-37*n^2+20*n-4)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^4 - 104*n^3 + 252*n^2 - 270*n + 108 + (24*n^5 - 180*n^4 + 520*n^3 - 728*n^2 + 496*n - 132)*x + (4*n^6 - 16*n^5 - 43*n^4 + 338*n^3 - 701*n^2 + 630*n - 214)*x^2 + (3*n^6 - 28*n^5 + 104*n^4 - 196*n^3 + 197*n^2 - 100*n + 20)*x^3)*P[n - 1, x] - ((8*n^3 - 36*n^2 + 54*n - 27)*x + (12*n^4 - 64*n^3 + 124*n^2 - 104*n + 32)*x^2 + (2*n^5 - n^4 - 40*n^3 + 123*n^2 - 138*n + 55)*x^3 + (2*n^5 - 13*n^4 + 32*n^3 - 37*n^2 + 20*n - 4)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 28*n^3 + 73*n^2 - 84*n + 36 + (6*n^5 - 60*n^4 + 238*n^3 - 468*n^2 + 456*n - 176)*x + (n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((16*n^4 - 104*n^3 + 252*n^2 - 270*n + 108 + (24*n^5 - 180*n^4 + 520*n^3 - 728*n^2 + 496*n - 132)*x + (4*n^6 - 16*n^5 - 43*n^4 + 338*n^3 - 701*n^2 + 630*n - 214)*x^2 + (3*n^6 - 28*n^5 + 104*n^4 - 196*n^3 + 197*n^2 - 100*n + 20)*x^3)*v[i - 1] - ((8*n^3 - 36*n^2 + 54*n - 27)*x + (12*n^4 - 64*n^3 + 124*n^2 - 104*n + 32)*x^2 + (2*n^5 - n^4 - 40*n^3 + 123*n^2 - 138*n + 55)*x^3 + (2*n^5 - 13*n^4 + 32*n^3 - 37*n^2 + 20*n - 4)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 28*n^3 + 73*n^2 - 84*n + 36 + (6*n^5 - 60*n^4 + 238*n^3 - 468*n^2 + 456*n - 176)*x + (n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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