Real-rooted atlas

A026374 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 9\,x + 30\,x^{2} + 45\,x^{3} + 30\,x^{4} + 9\,x^{5} + x^{6}\right)\,P_{n-6}(x) + \left(-3 - 18\,x - 33\,x^{2} - 18\,x^{3} - 3\,x^{4}\right)\,P_{n-4}(x) + \left(3 + 9\,x + 3\,x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 2\,x\)
\(P_{2}(x) = 7 + 12\,x + 3\,x^{2}\)
\(P_{3}(x) = 9 + 32\,x + 24\,x^{2} + 4\,x^{3}\)
\(P_{4}(x) = 13 + 78\,x + 105\,x^{2} + 44\,x^{3} + 5\,x^{4}\)
\(P_{5}(x) = 15 + 116\,x + 255\,x^{2} + 208\,x^{3} + 65\,x^{4} + 6\,x^{5}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
2712322
393224469
41378105445245
515116255208656665
6191986307804209672150
72125410171800154565412685425

All 15 rows

Technical details
Node id
n-5eddde11117f58a0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4d9e8666b18989c5

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,   2;
  2:  7,  12,    3;
  3:  9,  32,   24,    4;
  4: 13,  78,  105,   44,     5;
  5: 15, 116,  255,  208,    65,     6;
  6: 19, 198,  630,  780,   420,    96,    7;
  7: 21, 254, 1017, 1800,  1545,   654,  126,    8;
  8: 25, 372, 1902, 4476,  5295,  3300, 1078,  168,   9;
  9: 27, 446, 2634, 7588, 11865, 10518, 5334, 1496, 207, 10;
%F Formula
The row polynomials satisfy P(n,x) = (1+9*x+30*x^2+45*x^3+30*x^4+9*x^5+x^6)*P(n-6,x) - (3+18*x+33*x^2+18*x^3+3*x^4)*P(n-4,x) + (3+9*x+3*x^2)*P(n-2,x) for n >= 6, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 7 + 12*x + 3*x^2, P(3,x) = 9 + 32*x + 24*x^2 + 4*x^3, P(4,x) = 13 + 78*x + 105*x^2 + 44*x^3 + 5*x^4, P(5,x) = 15 + 116*x + 255*x^2 + 208*x^3 + 65*x^4 + 6*x^5.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[2, x_] := 7 + 12*x + 3*x^2;
P[3, x_] := 9 + 32*x + 24*x^2 + 4*x^3;
P[4, x_] := 13 + 78*x + 105*x^2 + 44*x^3 + 5*x^4;
P[5, x_] := 15 + 116*x + 255*x^2 + 208*x^3 + 65*x^4 + 6*x^5;
P[n_, x_] := P[n, x] = Expand[(1 + 9*x + 30*x^2 + 45*x^3 + 30*x^4 + 9*x^5 + x^6)*P[n - 6, x] - (3 + 18*x + 33*x^2 + 18*x^3 + 3*x^4)*P[n - 4, x] + (3 + 9*x + 3*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; v[3] = 7 + 12*x + 3*x^2; v[4] = 9 + 32*x + 24*x^2 + 4*x^3; v[5] = 13 + 78*x + 105*x^2 + 44*x^3 + 5*x^4; v[6] = 15 + 116*x + 255*x^2 + 208*x^3 + 65*x^4 + 6*x^5; for(n = 6, N, my(i = n + 1); v[i] = (1 + 9*x + 30*x^2 + 45*x^3 + 30*x^4 + 9*x^5 + x^6)*v[i - 6] - (3 + 18*x + 33*x^2 + 18*x^3 + 3*x^4)*v[i - 4] + (3 + 9*x + 3*x^2)*v[i - 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.