Real-rooted atlas

A200536 → D+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
147213
273249284120
31075222327246848972
413136602146421211848920224167344
5162151270433094001350312890796029605603253136
6193122307101402944559352848418619661260292808688134464373248
72242737942043374424193641370426527887561134440412249144971042396831361282566080

All 15 rows

Technical details
Node id
n-a7af5531257fc90d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d31e0778cbd986bd

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:  4,   7,    2;
  2:  7,  32,   49,    28,      4;
  3: 10,  75,  222,   327,    246,      84,       8;
  4: 13, 136,  602,  1464,   2121,    1848,     920,     224,       16;
  5: 16, 215, 1270,  4330,   9400,   13503,   12890,    7960,     2960,      560,       32;
  6: 19, 312, 2307, 10140,  29445,   59352,   84841,   86196,    61260,    29280,     8688,     1344,       64;
  7: 22, 427, 3794, 20433,  74424,  193641,  370426,  527887,   561134,   440412,   249144,    97104,    23968,    3136,     128;
  8: 25, 560, 5812, 37072, 162526,  518784, 1245748, 2292176,  3259921,  3586288,  3030832,  1936704,   909664,  299264,   63232,   7168,    256;
  9: 28, 711, 8442, 62244, 319176, 1207962, 3494148, 7891380, 14090508, 20011383, 22616370, 20241072, 14187936, 7642656, 3068352, 873216, 161280, 16128, 512;
%F Formula
The row polynomials satisfy (3*n-2+(4*n-1)*x+2*x^2)*P(n,x) = (3*n+1+(13*n+6)*x+(18*n+13)*x^2+(8*n+12)*x^3+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[((3*n + 1 + (13*n + 6)*x + (18*n + 13)*x^2 + (8*n + 12)*x^3 + 4*x^4)*P[n - 1, x])/(3*n - 2 + (4*n - 1)*x + 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] = ((3*n + 1 + (13*n + 6)*x + (18*n + 13)*x^2 + (8*n + 12)*x^3 + 4*x^4)*v[i - 1])/(3*n - 2 + (4*n - 1)*x + 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.