Real-rooted atlas

A075497 → θ+1 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2312419
3436568104
458030024016641
5615010402160992324380
672522800112001444840326432803
783926384420001088649273616256128266768
8957612936127008556080994560580800652802562337505
91081024000329280219139268040008730880358272026163251221925236
101111004158076032071665443446150478953600746240002188876810475521024218946003
1112145268200160380020355456140536704508282368882816000626177024132903936419225620482316939256
12131872106964314600051768288486300672256050995271587676169610881280518662144080371200016773120409625878593313
1314236616161658138081204403201478917440107071078404419903488097438268928102592089600425645875204847431680671006728192304020964876
14152940236600102211202603881284052048000386660102402218269081607324644687361292369734656107912663040034697994240029185241088268419072163843745210267939
Technical details
Node id
n-df7ae1bafc4127ea · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-ab7a3ee887f5ebf5

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:  3,  12,     4;
  3:  4,  36,    56,      8;
  4:  5,  80,   300,    240,      16;
  5:  6, 150,  1040,   2160,     992,      32;
  6:  7, 252,  2800,  11200,   14448,    4032,      64;
  7:  8, 392,  6384,  42000,  108864,   92736,   16256,     128;
  8:  9, 576, 12936, 127008,  556080,  994560,  580800,   65280,    256;
  9: 10, 810, 24000, 329280, 2191392, 6804000, 8730880, 3582720, 261632, 512;
%F Formula
The row polynomials satisfy P(n,x) = -(1+(2*n-4)*x)*P(n-2,x) + 2*x^2*d/dx P(n-2,x) + (2+2*n*x)*P(n-1,x) - 2*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(1 + (2*n - 4)*x)*P[n - 2, x] + 2*x^2*D[P[n - 2, x], x] + (2 + 2*n*x)*P[n - 1, 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; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + (2*n - 4)*x)*v[i - 2] + 2*x^2*deriv(v[i - 2], x) + (2 + 2*n*x)*v[i - 1] - 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. [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.