Real-rooted atlas

A128100 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(12\,n + 13\,n x + 9\,n^{2} x + 3\,n^{2} x^{2} + 3\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-48 - 160\,x - 156\,x^{2} - 36\,x^{3} + 60\,n + 119\,n x + 36\,n x^{2} - 18\,n x^{3} + 63\,n^{2} x + 111\,n^{2} x^{2} + 36\,n^{2} x^{3} + 15\,n^{3} x^{2} + 18\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-36 - 180\,x - 329\,x^{2} - 258\,x^{3} - 72\,x^{4} - 54\,n x - 207\,n x^{2} - 261\,n x^{3} - 108\,n x^{4} - 18\,n^{2} x^{2} - 51\,n^{2} x^{3} - 36\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2369
351217
411422780
5219681198
643246324108721
7855649184321999
81711314267319444056507
9341295270476480202518845
106836606183062106010125145858238
111365145564568462208384758748171036
12273131866112023177552137700481145103515089
13546169168268893484704447930204120357211515997
141092314928663633612879001389150796068214326174964501485
Technical details
Node id
n-8e9bf83669ab8f85 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d302b8af0fc2f5e8

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:   3,    6;
  3:   5,   12;
  4:  11,   42,   27;
  5:  21,   96,   81;
  6:  43,  246,  324,  108;
  7:  85,  564,  918,  432;
  8: 171, 1314, 2673, 1944,  405;
  9: 341, 2952, 7047, 6480, 2025;
%F Formula
The row polynomials satisfy (12*n+(9*n^2+13*n)*x+(3*n^3+3*n^2)*x^2)*P(n,x) = (60*n-48+(63*n^2+119*n-160)*x+(15*n^3+111*n^2+36*n-156)*x^2+(18*n^3+36*n^2-18*n-36)*x^3)*P(n-1,x) - (36+(54*n+180)*x+(18*n^2+207*n+329)*x^2+(51*n^2+261*n+258)*x^3+(36*n^2+108*n+72)*x^4)*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[((60*n - 48 + (63*n^2 + 119*n - 160)*x + (15*n^3 + 111*n^2 + 36*n - 156)*x^2 + (18*n^3 + 36*n^2 - 18*n - 36)*x^3)*P[n - 1, x] - (36 + (54*n + 180)*x + (18*n^2 + 207*n + 329)*x^2 + (51*n^2 + 261*n + 258)*x^3 + (36*n^2 + 108*n + 72)*x^4)*D[P[n - 1, x], x])/(12*n + (9*n^2 + 13*n)*x + (3*n^3 + 3*n^2)*x^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] = ((60*n - 48 + (63*n^2 + 119*n - 160)*x + (15*n^3 + 111*n^2 + 36*n - 156)*x^2 + (18*n^3 + 36*n^2 - 18*n - 36)*x^3)*v[i - 1] - (36 + (54*n + 180)*x + (18*n^2 + 207*n + 329)*x^2 + (51*n^2 + 261*n + 258)*x^3 + (36*n^2 + 108*n + 72)*x^4)*deriv(v[i - 1], x))/(12*n + (9*n^2 + 13*n)*x + (3*n^3 + 3*n^2)*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

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.