Real-rooted atlas

A008459 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n^{2} - 3\,n^{2} x + 6\,n^{3} x + n^{4} x^{2}\right)\,P_{n}(x) = \left(2\,x^{2} - 8\,n - 26\,n x - 22\,n x^{2} - 4\,n x^{3} + 16\,n^{2} + 24\,n^{2} x + 2\,n^{2} x^{2} - n^{2} x^{3} + 24\,n^{3} x + 28\,n^{3} x^{2} + 6\,n^{3} x^{3} + 4\,n^{4} x^{2} + 3\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(x^{2} + x^{3} - 8\,n x - 20\,n x^{2} - 16\,n x^{3} - 4\,n x^{4} - 12\,n^{2} x^{2} - 18\,n^{2} x^{3} - 6\,n^{2} x^{4} - 2\,n^{3} x^{3} - 2\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
2624939
32012010816264
470560810320251785
52522520504033607503611958
69241108828350268801050015124979303
734324804814968818480010500027216274464520992
8128702059207567561153152866250332640617404608813394017
9486208751603706560672672063063003243240905520126720729010021946230
1018475636951201772199037340160420420002724321610300290219648024057011000121140975703
1170543215519504831402001995364802625480002058376329888278428828800486486042900015972144900308808
122704156648997443841077241034633600155887875014366626568405036643136573447297290010010000726726224641695719779897
13104006002704156001752293088523524601688913825009428098680651819168029797447688931882961701700001937936011793603075819636169720902
14401166001123264800790965630025959897600490804314005889651768047053196190255406694409428098680232792560037063026035642880184548041160225227767934295
Technical details
Node id
n-6a86835eaaf82ea8 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e884231a3e2c9072

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:     2,      4;
  2:     6,     24,       9;
  3:    20,    120,     108,      16;
  4:    70,    560,     810,     320,      25;
  5:   252,   2520,    5040,    3360,     750,      36;
  6:   924,  11088,   28350,   26880,   10500,    1512,     49;
  7:  3432,  48048,  149688,  184800,  105000,   27216,   2744,     64;
  8: 12870, 205920,  756756, 1153152,  866250,  332640,  61740,   4608,   81;
  9: 48620, 875160, 3706560, 6726720, 6306300, 3243240, 905520, 126720, 7290, 100;
%F Formula
The row polynomials satisfy (4*n^2+(6*n^3-3*n^2)*x+n^4*x^2)*P(n,x) = (16*n^2-8*n+(24*n^3+24*n^2-26*n)*x+(4*n^4+28*n^3+2*n^2-22*n+2)*x^2+(3*n^4+6*n^3-n^2-4*n)*x^3)*P(n-1,x) - (8*n*x+(12*n^2+20*n-1)*x^2+(2*n^3+18*n^2+16*n-1)*x^3+(2*n^3+6*n^2+4*n)*x^4)*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[((16*n^2 - 8*n + (24*n^3 + 24*n^2 - 26*n)*x + (4*n^4 + 28*n^3 + 2*n^2 - 22*n + 2)*x^2 + (3*n^4 + 6*n^3 - n^2 - 4*n)*x^3)*P[n - 1, x] - (8*n*x + (12*n^2 + 20*n - 1)*x^2 + (2*n^3 + 18*n^2 + 16*n - 1)*x^3 + (2*n^3 + 6*n^2 + 4*n)*x^4)*D[P[n - 1, x], x])/(4*n^2 + (6*n^3 - 3*n^2)*x + n^4*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] = ((16*n^2 - 8*n + (24*n^3 + 24*n^2 - 26*n)*x + (4*n^4 + 28*n^3 + 2*n^2 - 22*n + 2)*x^2 + (3*n^4 + 6*n^3 - n^2 - 4*n)*x^3)*v[i - 1] - (8*n*x + (12*n^2 + 20*n - 1)*x^2 + (2*n^3 + 18*n^2 + 16*n - 1)*x^3 + (2*n^3 + 6*n^2 + 4*n)*x^4)*deriv(v[i - 1], x))/(4*n^2 + (6*n^3 - 3*n^2)*x + n^4*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

Same fingerprint ?

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