Real-rooted atlas

basis-map coefficients of A026729 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-7 + 4\,x + 4\,n\right)\,P_{n}(x) = \left(-3 - 8\,x + 16\,x^{2} + 4\,n + 16\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2549
39401665
41310824064425
51720886412802562625
621340208057606400102415625
7255044080166403456030720409690625
82970070563808011648019353614336016384515625

All 15 rows

Technical details
Node id
n-c41e614167a72a8f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-236e6d52004bfa9f

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:  5,    4;
   3:  9,   40,    16;
   4: 13,  108,   240,     64;
   5: 17,  208,   864,   1280,    256;
   6: 21,  340,  2080,   5760,   6400,    1024;
   7: 25,  504,  4080,  16640,  34560,   30720,    4096;
   8: 29,  700,  7056,  38080, 116480,  193536,  143360,   16384;
   9: 33,  928, 11200,  75264, 304640,  745472, 1032192,  655360,   65536;
  10: 37, 1188, 16704, 134400, 677376, 2193408, 4472832, 5308416, 2949120, 262144;
%F Formula
The row polynomials satisfy (4*n-7+4*x)*P(n,x) = (4*n-3+(16*n-8)*x+16*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[((4*n - 3 + (16*n - 8)*x + 16*x^2)*P[n - 1, x])/(4*n - 7 + 4*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] = ((4*n - 3 + (16*n - 8)*x + 16*x^2)*v[i - 1])/(4*n - 7 + 4*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.