Real-rooted atlas

A386789 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2915630
3441048020248
4225690780385702150
51182445866364872176425219164
66321283015222050820275107854924173950
7342321774403905284730403405601457283432034321599280
8187137110221228173604081770366597020900887387381480051287014844150
9103049067975701978724033359480358958602556554012052040362648063206048620138795380
10570744941682267136113714261712308328113786280269990165189024663371281737192626789621847561305361310
113176067625439997692090232019885811402846156040283497177620053575361007388720352212120816285601128691270543212335351208
1217743529715466561506146648508147265468352369000634026956561632222480258001341938254858710837901817075260377641264473227302704156117027090310
13994551222937225972240565950740106778936880190704701880244035309672230194833792162065405040851617859403298939182091571042641724835504197611400104006001113953078476
14559040278556633660505265179315960760667352120149335683354021226108761002252583950760181317136572011124117189905176710696301798765940404525598220077937639008223903004011660010633665393150

Other recurrences

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

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(2\,n - n^{2}\right)\,P_{n-2}(x) + \left(3 + 2\,x - 9\,n - 6\,n x + 6\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)
Technical details
Node id
n-864f996c2a017221 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-f4e0dab53593f1d4)
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:       9,      15,        6;
  3:      44,     104,       80,       20;
  4:     225,     690,      780,      385,       70;
  5:    1182,    4458,     6636,     4872,     1764,      252;
  6:    6321,   28301,    52220,    50820,    27510,     7854,      924;
  7:   34232,  177440,   390528,   473040,   340560,   145728,    34320,    3432;
  8:  187137, 1102212,  2817360,  4081770,  3665970,  2090088,   738738,  148005,  12870;
  9: 1030490, 6797570, 19787240, 33359480, 35895860, 25565540, 12052040, 3626480, 632060, 48620;
%F Formula
The row polynomials satisfy n*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[Together[((3*n - 1 + 2*n*x)*P[n - 1, x] + (4 + 6*x + 2*x^2)*D[P[n - 1, x], x])/(n)]];
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))/(n)); 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.