Real-rooted atlas

A125662 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(36\,n + 6\,n x + 4\,n x^{2} - 30\,n^{3} x^{2} - 10\,n^{3} x^{3} + n^{5} x^{4}\right)\,P_{n}(x) = \left(72 + 93\,x + 38\,x^{2} - 2\,x^{3} - 4\,x^{4} + 72\,n + 156\,n x + 32\,n x^{2} - 53\,n x^{3} - 22\,n x^{4} + 63\,n^{2} x - 66\,n^{2} x^{2} - 116\,n^{2} x^{3} - 36\,n^{2} x^{4} + n^{2} x^{5} - 60\,n^{3} x^{2} - 71\,n^{3} x^{3} - 18\,n^{3} x^{4} + 3\,n^{3} x^{5} - 6\,n^{4} x^{3} + 2\,n^{4} x^{4} + 3\,n^{4} x^{5} + 2\,n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x) + \left(27 + 54\,x + 27\,x^{2} - 8\,x^{3} - 10\,x^{4} - 2\,x^{5} - 63\,n x^{2} - 114\,n x^{3} - 61\,n x^{4} - 10\,n x^{5} - 63\,n^{2} x^{2} - 114\,n^{2} x^{3} - 58\,n^{2} x^{4} - 6\,n^{2} x^{5} + n^{2} x^{6} + 6\,n^{3} x^{4} + 8\,n^{3} x^{5} + 2\,n^{3} x^{6} + 3\,n^{4} x^{4} + 4\,n^{4} x^{5} + n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
2156436115
356368432128984
42091856334820484007861
5780872821312199688000115259940
629113929612117615360094000276483136441767
71086417174464022410252808480003801608780881923172272

All 15 rows

Technical details
Node id
n-d9bba24ba2c6a274 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-a9850b8ef7702cbe

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:      4,       8;
  2:     15,      64,       36;
  3:     56,     368,      432,      128;
  4:    209,    1856,     3348,     2048,      400;
  5:    780,    8728,    21312,    19968,     8000,     1152;
  6:   2911,   39296,   121176,   153600,    94000,    27648,     3136;
  7:  10864,  171744,   640224,  1025280,   848000,   380160,    87808,    8192;
  8:  40545,  734592,  3212568,  6223872,  6502000,  3935232,  1382976,  262144,  20736;
  9: 151316, 3090984, 15515712, 35292672, 44609600, 34086528, 16156672, 4653056, 746496, 51200;
%F Formula
The row polynomials satisfy (36*n+6*n*x-(30*n^3-4*n)*x^2-10*n^3*x^3+n^5*x^4)*P(n,x) = (72*n+72+(63*n^2+156*n+93)*x-(60*n^3+66*n^2-32*n-38)*x^2-(6*n^4+71*n^3+116*n^2+53*n+2)*x^3+(2*n^5+2*n^4-18*n^3-36*n^2-22*n-4)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (27+54*x-(63*n^2+63*n-27)*x^2-(114*n^2+114*n+8)*x^3+(3*n^4+6*n^3-58*n^2-61*n-10)*x^4+(4*n^4+8*n^3-6*n^2-10*n-2)*x^5+(n^4+2*n^3+n^2)*x^6)*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[((72*n + 72 + (63*n^2 + 156*n + 93)*x - (60*n^3 + 66*n^2 - 32*n - 38)*x^2 - (6*n^4 + 71*n^3 + 116*n^2 + 53*n + 2)*x^3 + (2*n^5 + 2*n^4 - 18*n^3 - 36*n^2 - 22*n - 4)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (27 + 54*x - (63*n^2 + 63*n - 27)*x^2 - (114*n^2 + 114*n + 8)*x^3 + (3*n^4 + 6*n^3 - 58*n^2 - 61*n - 10)*x^4 + (4*n^4 + 8*n^3 - 6*n^2 - 10*n - 2)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(36*n + 6*n*x - (30*n^3 - 4*n)*x^2 - 10*n^3*x^3 + n^5*x^4)]];
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] = ((72*n + 72 + (63*n^2 + 156*n + 93)*x - (60*n^3 + 66*n^2 - 32*n - 38)*x^2 - (6*n^4 + 71*n^3 + 116*n^2 + 53*n + 2)*x^3 + (2*n^5 + 2*n^4 - 18*n^3 - 36*n^2 - 22*n - 4)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (27 + 54*x - (63*n^2 + 63*n - 27)*x^2 - (114*n^2 + 114*n + 8)*x^3 + (3*n^4 + 6*n^3 - 58*n^2 - 61*n - 10)*x^4 + (4*n^4 + 8*n^3 - 6*n^2 - 10*n - 2)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(36*n + 6*n*x - (30*n^3 - 4*n)*x^2 - 10*n^3*x^3 + n^5*x^4)); 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