Real-rooted atlas

A114687 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,x + n + 6\,n x + 2\,n x^{2} + 2\,n^{2} x + 3\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(2 + 4\,x + 7\,n + 28\,n x + 26\,n x^{2} + 4\,n x^{3} + 16\,n^{2} x + 33\,n^{2} x^{2} + 14\,n^{2} x^{3} + 7\,n^{3} x^{2} + 6\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-1 - 7\,x - 16\,x^{2} - 14\,x^{3} - 4\,x^{4} - 4\,n x - 17\,n x^{2} - 23\,n x^{3} - 10\,n x^{4} - 2\,n^{2} x^{2} - 6\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
211281251
34516814432389
41979681248576803069
5903550094806720200019224795
64279310926714064480300006336448203775
7207931755044556165532803528001182721881610241696105

All 15 rows

Technical details
Node id
n-88365a146af6ec86 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9718b2eeff65c407

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,       4;
  2:     11,      28,       12;
  3:     45,     168,      144,       32;
  4:    197,     968,     1248,      576,       80;
  5:    903,    5500,     9480,     6720,     2000,      192;
  6:   4279,   31092,    67140,    64480,    30000,     6336,     448;
  7:  20793,  175504,   455616,   553280,   352800,   118272,   18816,    1024;
  8: 103049,  990736,  3005184,  4417280,  3590720,  1666560,  426496,   53248,   2304;
  9: 518859, 5597172, 19427472, 33540864, 33203520, 19789056, 7074816, 1437696, 145152, 5120;
%F Formula
The row polynomials satisfy (n+2+(2*n^2+6*n+4)*x+(n^3+3*n^2+2*n)*x^2)*P(n,x) = (7*n+2+(16*n^2+28*n+4)*x+(7*n^3+33*n^2+26*n)*x^2+(6*n^3+14*n^2+4*n)*x^3)*P(n-1,x) - (1+(4*n+7)*x+(2*n^2+17*n+16)*x^2+(6*n^2+23*n+14)*x^3+(4*n^2+10*n+4)*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[((7*n + 2 + (16*n^2 + 28*n + 4)*x + (7*n^3 + 33*n^2 + 26*n)*x^2 + (6*n^3 + 14*n^2 + 4*n)*x^3)*P[n - 1, x] - (1 + (4*n + 7)*x + (2*n^2 + 17*n + 16)*x^2 + (6*n^2 + 23*n + 14)*x^3 + (4*n^2 + 10*n + 4)*x^4)*D[P[n - 1, x], x])/(n + 2 + (2*n^2 + 6*n + 4)*x + (n^3 + 3*n^2 + 2*n)*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] = ((7*n + 2 + (16*n^2 + 28*n + 4)*x + (7*n^3 + 33*n^2 + 26*n)*x^2 + (6*n^3 + 14*n^2 + 4*n)*x^3)*v[i - 1] - (1 + (4*n + 7)*x + (2*n^2 + 17*n + 16)*x^2 + (6*n^2 + 23*n + 14)*x^3 + (4*n^2 + 10*n + 4)*x^4)*deriv(v[i - 1], x))/(n + 2 + (2*n^2 + 6*n + 4)*x + (n^3 + 3*n^2 + 2*n)*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