Real-rooted atlas

A227450 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
022
1628
22216240
394110342240
44547483926821664
52430522639641236134212992
61421438008382781779236982642112256
789918288998366062229974735461081852221059840
86101822299204352943628220841242872291052313961036210827264
9441279819125602346334323384498419205860631703611255609106420622118758400
1033827974166117136347662922403313920282153148121115872309477324303616264926411221389711360
1127364652615040703663580195898482355114640316977402152733260727517652148315636163826787741988210217258893312
1223269809981417221090837876079568582635771885683848045436499409048154859586724234044488701845042623518202273312164042226463227904
132073250406213873733564241191267458071365514812479800808981860112761729430881483698410702099485624141255506329943663023786027667050843279023127694491648
1419298272935014087524066964605572013934888747121692811228633261438973395758004058962609395902491430641536719452034322135870057016154719861229108171201985290665560245316785602560
Technical details
Node id
n-df199653ca102fb5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-5344ee972d4f418a

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:       2;
  1:       6,        2;
  2:      22,       16,        2;
  3:      94,      110,       34,        2;
  4:     454,      748,      392,       68,        2;
  5:    2430,     5226,     3964,     1236,      134,       2;
  6:   14214,    38008,    38278,    17792,     3698,     264,       2;
  7:   89918,   288998,   366062,   229974,    73546,   10818,     522,     2;
  8:  610182,  2299204,  3529436,  2822084,  1242872,  291052,   31396,  1036,    2;
  9: 4412798, 19125602, 34633432, 33844984, 19205860, 6317036, 1125560, 91064, 2062, 2;
%F Formula
The row polynomials satisfy P(n,x) = (n+2+n*x)*P(n-1,x) - (1+2*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.
%t Mathematica
P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(n + 2 + n*x)*P[n - 1, x] - (1 + 2*x + 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] = 2; for(n = 1, N, my(i = n + 1); v[i] = (n + 2 + n*x)*v[i - 1] - (1 + 2*x + 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. [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.