Real-rooted atlas

A196347 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + 3\,x + 3\,n\right)\,P_{n}(x) = \left(4\,n + 12\,n x + 9\,n x^{2} + 6\,n^{2} + 9\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1538
2326018110
32648648101622100
42688126722073612960194451000
532640201600475200518400243000291601500000
64608003525120108864001710720013996800524880052488051750000
7741888067737600259096320533433600628689600411505920128595600110224802047500000

All 15 rows

Technical details
Node id
n-6ae06261a6a6ad06 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-bff12ef9596281bf

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:          5,           3;
  2:         32,          60,           18;
  3:        264,         864,          810,          162;
  4:       2688,       12672,        20736,        12960,          1944;
  5:      32640,      201600,       475200,       518400,        243000,         29160;
  6:     460800,     3525120,     10886400,     17107200,      13996800,       5248800,       524880;
  7:    7418880,    67737600,    259096320,    533433600,     628689600,     411505920,    128595600,     11022480;
  8:  134184960,  1424424960,   6502809600,  16582164480,   25604812800,   24141680640,  13168189440,   3527193600,    264539520;
  9: 2694021120, 32606945280, 173067632640, 526727577600, 1007366492160, 1244393902080, 977738065920, 457124290560, 107138505600, 7142567040;
%F Formula
The row polynomials satisfy (3*n-1+3*x)*P(n,x) = (6*n^2+4*n+(9*n^2+12*n)*x+9*n*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[((6*n^2 + 4*n + (9*n^2 + 12*n)*x + 9*n*x^2)*P[n - 1, x])/(3*n - 1 + 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] = ((6*n^2 + 4*n + (9*n^2 + 12*n)*x + 9*n*x^2)*v[i - 1])/(3*n - 1 + 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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.