Real-rooted atlas

A357367 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
218301260
32646244801201488
4540016560187209240168051600
5141840534960796320584640211680302402299680
64551120203767203759840036590400198072005654880665280125244000
7172529280894297600196826112023841216001716422400734469120172972800172972808060371200

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(2\,n - n^{2}\right)\,P_{n-2}(x) + \left(-3 - 2\,x + 6\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)
Technical details
Node id
n-79beb834849e7646 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-d6bf80c66f6fc720)
Fingerprint
f6-8f228763548ec508

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:            2,             2;
  2:           18,            30,            12;
  3:          264,           624,           480,            120;
  4:         5400,         16560,         18720,           9240,           1680;
  5:       141840,        534960,        796320,         584640,         211680,         30240;
  6:      4551120,      20376720,      37598400,       36590400,       19807200,       5654880,        665280;
  7:    172529280,     894297600,    1968261120,     2384121600,     1716422400,     734469120,     172972800,      17297280;
  8:   7545363840,   44441187840,  113595955200,   164576966400,   147811910400,   84272348160,   29785916160,    5967561600,    518918400;
  9: 373944211200, 2466702201600, 7180393651200, 12105488102400, 13025889676800, 9277223155200, 4373444275200, 1315977062400, 229361932800, 17643225600;
%F Formula
The row polynomials satisfy P(n,x) = (3*n-1+2*n*x)*P(n-1,x) + (4+6*x+2*x^2)*d/dx 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[(3*n - 1 + 2*n*x)*P[n - 1, x] + (4 + 6*x + 2*x^2)*D[P[n - 1, x], 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 + 2*n*x)*v[i - 1] + (4 + 6*x + 2*x^2)*deriv(v[i - 1], 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.