Real-rooted atlas

A026374 → D+1 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 9\,x + 30\,x^{2} + 45\,x^{3} + 30\,x^{4} + 9\,x^{5} + x^{6}\right)\,P_{n-6}(x) + \left(-3 - 18\,x - 33\,x^{2} - 18\,x^{3} - 3\,x^{4}\right)\,P_{n-4}(x) + \left(3 + 9\,x + 3\,x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)
\(P_{2}(x) = 4 + 10\,x + 3\,x^{2}\)
\(P_{3}(x) = 5 + 24\,x + 21\,x^{2} + 4\,x^{3}\)
\(P_{4}(x) = 7 + 56\,x + 87\,x^{2} + 40\,x^{3} + 5\,x^{4}\)
\(P_{5}(x) = 8 + 82\,x + 204\,x^{2} + 180\,x^{3} + 60\,x^{4} + 6\,x^{5}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2410317
352421454
475687405195
5882204180606540
6101384956603759071775
7111767921500135059411984550

All 15 rows

Technical details
Node id
n-ef68c61660e41606 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-a25c0a0957a3bd5b

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,   2;
  2:  4,  10,    3;
  3:  5,  24,   21,    4;
  4:  7,  56,   87,   40,     5;
  5:  8,  82,  204,  180,    60,    6;
  6: 10, 138,  495,  660,   375,   90,    7;
  7: 11, 176,  792, 1500,  1350,  594,  119,    8;
  8: 13, 256, 1470, 3696,  4575, 2952,  994,  160,   9;
  9: 14, 306, 2028, 6232, 10170, 9306, 4844, 1392, 198, 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) = 2 + 2*x, P(2,x) = 4 + 10*x + 3*x^2, P(3,x) = 5 + 24*x + 21*x^2 + 4*x^3, P(4,x) = 7 + 56*x + 87*x^2 + 40*x^3 + 5*x^4, P(5,x) = 8 + 82*x + 204*x^2 + 180*x^3 + 60*x^4 + 6*x^5.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 4 + 10*x + 3*x^2;
P[3, x_] := 5 + 24*x + 21*x^2 + 4*x^3;
P[4, x_] := 7 + 56*x + 87*x^2 + 40*x^3 + 5*x^4;
P[5, x_] := 8 + 82*x + 204*x^2 + 180*x^3 + 60*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] = 2 + 2*x; v[3] = 4 + 10*x + 3*x^2; v[4] = 5 + 24*x + 21*x^2 + 4*x^3; v[5] = 7 + 56*x + 87*x^2 + 40*x^3 + 5*x^4; v[6] = 8 + 82*x + 204*x^2 + 180*x^3 + 60*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.