Real-rooted atlas

A080248 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
21416
311013125
412073401135
51352734781211909
61567983208298936417417
718419741517835069182981093171698
8112043265663226273936898811088132801806968
9116586461777601452011431007338007616685669841110427825
1012201607148901664292913480230468451856386761764021237295241152915697
111286281711212211240338672148224527993002401065454016390782997241569468857312519879761
121364470472761616785833621080041664681432889617850759056163725932373931986916145030249265720146276398129
131455754395866718230472242461629295445695828800208651968144391238533413249520885471394648994098704472147971611940296067422
1415601168441175096061767563017292266264253429011730185370180494462334936414458465530087144378227818147439551944130452234804452391484121007099850230
Technical details
Node id
n-cf79907eb588e544 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-8583efd7a1bc8a2a

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: 1,   1;
  2: 1,   4,    1;
  3: 1,  10,   13,      1;
  4: 1,  20,   73,     40,       1;
  5: 1,  35,  273,    478,     121,       1;
  6: 1,  56,  798,   3208,    2989,     364,       1;
  7: 1,  84, 1974,  15178,   35069,   18298,    1093,      1;
  8: 1, 120, 4326,  56632,  262739,  368988,  110881,   3280,    1;
  9: 1, 165, 8646, 177760, 1452011, 4310073, 3800761, 668566, 9841, 1;
%F Formula
The row polynomials satisfy 2*P(n,x) = (2+(n^2+n)*x)*P(n-1,x) - 2*n*x^2*d/dx P(n-1,x) + x^3*d^2/dx^2 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[((2 + (n^2 + n)*x)*P[n - 1, x] - 2*n*x^2*D[P[n - 1, x], x] + x^3*D[P[n - 1, x], {x, 2}])/(2)]];
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] = ((2 + (n^2 + n)*x)*v[i - 1] - 2*n*x^2*deriv(v[i - 1], x) + x^3*deriv(deriv(v[i - 1], x), x))/(2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.