Real-rooted atlas

A026374 → D+1 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 9\,x + 30\,x^{2} + 45\,x^{3} + 30\,x^{4} + 9\,x^{5} + x^{6}\right)\,P_{n-6}(x) + \left(-3 - 18\,x - 33\,x^{2} - 18\,x^{3} - 3\,x^{4}\right)\,P_{n-4}(x) + \left(3 + 9\,x + 3\,x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)
\(P_{2}(x) = 9 + 7\,x + x^{2}\)
\(P_{3}(x) = 17 + 26\,x + 10\,x^{2} + x^{3}\)
\(P_{4}(x) = 35 + 86\,x + 59\,x^{2} + 14\,x^{3} + x^{4}\)
\(P_{5}(x) = 49 + 177\,x + 203\,x^{2} + 93\,x^{3} + 17\,x^{4} + x^{5}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
297117
3172610154
4358659141195
54917720393171540
6793996604651502111775
799616138914557652012414550

All 15 rows

Technical details
Node id
n-613fd6d4c10714ae · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-39cf3faa0042100c

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,    1;
  2:   9,    7,    1;
  3:  17,   26,   10,    1;
  4:  35,   86,   59,   14,    1;
  5:  49,  177,  203,   93,   17,    1;
  6:  79,  399,  660,  465,  150,   21,    1;
  7:  99,  616, 1389, 1455,  765,  201,   24,   1;
  8: 141, 1108, 3262, 4584, 3375, 1344,  282,  28,  1;
  9: 167, 1505, 5350, 9694, 9789, 5703, 1910, 350, 31, 1;
%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 + x, P(2,x) = 9 + 7*x + x^2, P(3,x) = 17 + 26*x + 10*x^2 + x^3, P(4,x) = 35 + 86*x + 59*x^2 + 14*x^3 + x^4, P(5,x) = 49 + 177*x + 203*x^2 + 93*x^3 + 17*x^4 + x^5.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[2, x_] := 9 + 7*x + x^2;
P[3, x_] := 17 + 26*x + 10*x^2 + x^3;
P[4, x_] := 35 + 86*x + 59*x^2 + 14*x^3 + x^4;
P[5, x_] := 49 + 177*x + 203*x^2 + 93*x^3 + 17*x^4 + 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 + x; v[3] = 9 + 7*x + x^2; v[4] = 17 + 26*x + 10*x^2 + x^3; v[5] = 35 + 86*x + 59*x^2 + 14*x^3 + x^4; v[6] = 49 + 177*x + 203*x^2 + 93*x^3 + 17*x^4 + 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.