Real-rooted atlas

basis-map coefficients of A030528, shifted → basis A013610 → D+1

only rational roots

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

Recurrence

\(\left(-2 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(-1 - 6\,x - 9\,x^{2} - 3\,n - 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1-4-3-7
2724940
3-10-63-108-27-208
413120378432811024
5-16-195-900-1890-1620-243-4864
619288175554008505583272922528
7-22-399-3024-12285-28350-35721-20412-2187-102400
82552847882419273710136080142884699846561458752
9-28-675-7128-43092-163296-398034-612360-551124-236196-19683-2031616
10318401012571280323190979776199017026244002066715787320590498912896
11-34-1023-13860-111375-588060-2133054-5388768-9382230-10825650-7577955-2598156-177147-38797312
12371224184141663201002375423403212798324277136644222003543302600272806388503056531441167772160
13-40-1443-23868-239382-1621620-7818525-27521208-71304948-135104112-182953485-168880140-96722262-27634932-1594323-721420288
1443168030303334152251351113621608547296751651272483743509776304858567684046376448150803385279178928208847829693087007744
Technical details
Node id
n-1c98e5590738cedf · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1f45c55ca90e4ac8

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): no
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:  -4,   -3;
  2:   7,   24,     9;
  3: -10,  -63,  -108,    -27;
  4:  13,  120,   378,    432,      81;
  5: -16, -195,  -900,  -1890,   -1620,    -243;
  6:  19,  288,  1755,   5400,    8505,    5832,     729;
  7: -22, -399, -3024, -12285,  -28350,  -35721,  -20412,   -2187;
  8:  25,  528,  4788,  24192,   73710,  136080,  142884,   69984,    6561;
  9: -28, -675, -7128, -43092, -163296, -398034, -612360, -551124, -236196, -19683;
%F Formula
The row polynomials satisfy (3*n-2+3*x)*P(n,x) = -(3*n+1+(9*n+6)*x+9*x^2)*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 + (9*n + 6)*x + 9*x^2)*P[n - 1, x])/(3*n - 2 + 3*x)]];
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 + (9*n + 6)*x + 9*x^2)*v[i - 1])/(3*n - 2 + 3*x)); 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

Same fingerprint ?

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.