Real-rooted atlas

A019538 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
222
361218
4147272158
5303007204801530
662108046807200360016622
7126361225200672007560030240201978
8254115921224725040009576008467202822402724878
951036300559440333648095256001397088010160640290304040492890
1010221119602455560204120008217720017781120021168000013063680032659200657944942
11204634201210494000118430400646153200193496688033530112003353011200179625600039916800011611834938
124094103831244028072662112000476514720018972455040448837804806466521600055683936000263450880005269017600221291822798
138190313950018234216036040804803355672320017283442176053614649088010436486860801284323040000970377408000410983372800747242496004530383962650
141638294676407482056401923601680022837094280014917174272005902293196800149074877952002458427811840026371433088000177407822592006799906713600113331778560099179581033262
153276628501212304962840010116812160015144957828001236033790992061141041561600194876691609600413552018880000590560274304000560992303872000339995335680000118998367488000183074411520002312402554523898
Technical details
Node id
n-39021415b377ba8d · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-18b3ed592c581b89

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    0;
   2:    2;
   3:    6,     12;
   4:   14,     72,      72;
   5:   30,    300,     720,      480;
   6:   62,   1080,    4680,     7200,     3600;
   7:  126,   3612,   25200,    67200,    75600,     30240;
   8:  254,  11592,  122472,   504000,   957600,    846720,    282240;
   9:  510,  36300,  559440,  3336480,  9525600,  13970880,  10160640,   2903040;
  10: 1022, 111960, 2455560, 20412000, 82177200, 177811200, 211680000, 130636800, 32659200;
%F Formula
The row polynomials satisfy P(n,x) = -(2+6*x+6*x^2)*P(n-2,x) - (x+3*x^2+2*x^3)*d/dx P(n-2,x) + (3+6*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(2 + 6*x + 6*x^2)*P[n - 2, x] - (x + 3*x^2 + 2*x^3)*D[P[n - 2, x], x] + (3 + 6*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = -(2 + 6*x + 6*x^2)*v[i - 2] - (x + 3*x^2 + 2*x^3)*deriv(v[i - 2], x) + (3 + 6*x)*v[i - 1] + (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, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.