Real-rooted atlas

A181543 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(6 + 6\,x - 15\,n - 15\,n x + 11\,n^{2} + 10\,n^{2} x\right)\,P_{n-1}(x) + \left(24 + 42\,x + 18\,x^{2} - 12\,n - 27\,n x - 15\,n x^{2}\right)\,P_{n-1}'(x) + \left(12 + 30\,x + 24\,x^{2} + 6\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
21010121
35684301171
43466924146811521
52252563047601510130114283
61518445552498002368044702221138909
7104960367360492912313880943601134035011385163

All 15 rows

Technical details
Node id
n-3397151940607f72 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e0dcd909a84c02c1

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,        1;
  2:      10,       10,        1;
  3:      56,       84,       30,        1;
  4:     346,      692,      414,       68,        1;
  5:    2252,     5630,     4760,     1510,      130,       1;
  6:   15184,    45552,    49800,    23680,     4470,     222,      1;
  7:  104960,   367360,   492912,   313880,    94360,   11340,    350,     1;
  8:  739162,  2956648,  4703020,  3760792,  1568350,  318136,  25564,   520,   1;
  9: 5280932, 23764194, 43721568, 42125916, 22576932, 6577326, 939792, 52524, 738, 1;
%F Formula
The row polynomials satisfy n^2*P(n,x) = (11*n^2-15*n+6+(10*n^2-15*n+6)*x)*P(n-1,x) - (12*n-24+(27*n-42)*x+(15*n-18)*x^2)*d/dx P(n-1,x) + (12+30*x+24*x^2+6*x^3)*d^2/dx^2 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[((11*n^2 - 15*n + 6 + (10*n^2 - 15*n + 6)*x)*P[n - 1, x] - (12*n - 24 + (27*n - 42)*x + (15*n - 18)*x^2)*D[P[n - 1, x], x] + (12 + 30*x + 24*x^2 + 6*x^3)*D[P[n - 1, x], {x, 2}])/(n^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] = ((11*n^2 - 15*n + 6 + (10*n^2 - 15*n + 6)*x)*v[i - 1] - (12*n - 24 + (27*n - 42)*x + (15*n - 18)*x^2)*deriv(v[i - 1], x) + (12 + 30*x + 24*x^2 + 6*x^3)*deriv(deriv(v[i - 1], x), x))/(n^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