Real-rooted atlas

basis-map coefficients of A026729 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-5 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(-2 - 3\,x + 9\,x^{2} + 3\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2437
3724940
4106310827208
513120378432811024
616195900189016202434864
719288175554008505583272922528
8223993024122852835035721204122187102400
92552847882419273710136080142884699846561458752
1028675712843092163296398034612360551124236196196832031616
11318401012571280323190979776199017026244002066715787320590498912896
1234102313860111375588060213305453887689382230108256507577955259815617714738797312
13371224184141663201002375423403212798324277136644222003543302600272806388503056531441167772160
144014432386823938216216207818525275212087130494813510411218295348516888014096722262276349321594323721420288
1543168030303334152251351113621608547296751651272483743509776304858567684046376448150803385279178928208847829693087007744
Technical details
Node id
n-33b5464f4fbbfa66 · 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 -1)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  4,   3;
   3:  7,  24,    9;
   4: 10,  63,  108,    27;
   5: 13, 120,  378,   432,     81;
   6: 16, 195,  900,  1890,   1620,    243;
   7: 19, 288, 1755,  5400,   8505,   5832,    729;
   8: 22, 399, 3024, 12285,  28350,  35721,  20412,   2187;
   9: 25, 528, 4788, 24192,  73710, 136080, 142884,  69984,   6561;
  10: 28, 675, 7128, 43092, 163296, 398034, 612360, 551124, 236196, 19683;
%F Formula
The row polynomials satisfy (3*n-5+3*x)*P(n,x) = (3*n-2+(9*n-3)*x+9*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 2 + (9*n - 3)*x + 9*x^2)*P[n - 1, x])/(3*n - 5 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((3*n - 2 + (9*n - 3)*x + 9*x^2)*v[i - 1])/(3*n - 5 + 3*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.