Real-rooted atlas

A200536 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 - 2\,x^{2} + 3\,n x + 4\,n x^{2}\right)\,P_{n}(x) = \left(1 + 6\,x + 13\,x^{2} + 12\,x^{3} + 4\,x^{4} + 3\,n x + 13\,n x^{2} + 18\,n x^{3} + 8\,n x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
116613
2112394820120
31189925233021656972
41241867201605216017367681447344
5130300156049251009813790124807200240035253136
61364412880117753250862923866888478057600258726912832373248
714260947882404583538209671389112539154556920423192229824844481881619202566080

All 15 rows

Technical details
Node id
n-23fe4c81851902da · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9d74e40d8b613922

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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: 1,  6,    6;
  2: 1, 12,   39,    48,    20;
  3: 1, 18,   99,   252,   330,    216,      56;
  4: 1, 24,  186,   720,  1605,   2160,    1736,     768,     144;
  5: 1, 30,  300,  1560,  4925,  10098,   13790,   12480,    7200,     2400,      352;
  6: 1, 36,  441,  2880, 11775,  32508,   62923,   86688,   84780,    57600,    25872,     6912,      832;
  7: 1, 42,  609,  4788, 24045,  83538,  209671,  389112,  539154,   556920,   423192,   229824,    84448,    18816,    1920;
  8: 1, 48,  804,  7392, 44030, 184464,  569380, 1330752, 2391561,  3326880,  3578960,  2951424,  1831648,   827904,  257280,   49152,   4352;
  9: 1, 54, 1026, 10800, 74430, 365148, 1338288, 3774816, 8346969, 14625630, 20403702, 22648896, 19883136, 13632192, 7145280, 2764800, 744192, 124416, 9728;
%F Formula
The row polynomials satisfy (1+3*n*x+(4*n-2)*x^2)*P(n,x) = (1+(3*n+6)*x+(13*n+13)*x^2+(18*n+12)*x^3+(8*n+4)*x^4)*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[((1 + (3*n + 6)*x + (13*n + 13)*x^2 + (18*n + 12)*x^3 + (8*n + 4)*x^4)*P[n - 1, x])/(1 + 3*n*x + (4*n - 2)*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] = ((1 + (3*n + 6)*x + (13*n + 13)*x^2 + (18*n + 12)*x^3 + (8*n + 4)*x^4)*v[i - 1])/(1 + 3*n*x + (4*n - 2)*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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.