Real-rooted atlas

A249139 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(26 + 32\,x + 6\,x^{2} + 26\,n + 45\,n x + 15\,n x^{2} + 13\,n^{2} x + 12\,n^{2} x^{2} + 3\,n^{3} x^{2}\right)\,P_{n}(x) = \left(26 + 32\,x + 6\,x^{2} + 182\,n + 510\,n x + 435\,n x^{2} + 108\,n x^{3} + 130\,n^{2} x + 222\,n^{2} x^{2} + 90\,n^{2} x^{3} + 21\,n^{3} x^{2} + 18\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-169 - 741\,x - 1190\,x^{2} - 834\,x^{3} - 216\,x^{4} - 169\,n x - 520\,n x^{2} - 531\,n x^{3} - 180\,n x^{4} - 39\,n^{2} x^{2} - 75\,n^{2} x^{3} - 36\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
271219
3195427100
44013281253
5974084051081018
6217104412424322935
750828504293237640510432
811597284127178640202531825
92683188823842132940121501458106534
10616047688109350109728506258748332299
11142091204323095823622322106005686251031079020
12326893004568522281128816773955265356357213389221
137531674700623229183453840276412511941022500471749610824850
14173383184450862316541024920093190504776408128595613996834020127
Technical details
Node id
n-b569f4a0f2c54d5e · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8126c1326009a233

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half affine: slope 1, intercept 1)
  • 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:    4,     6;
  2:    7,    12;
  3:   19,    54,    27;
  4:   40,   132,    81;
  5:   97,   408,   405,   108;
  6:  217,  1044,  1242,   432;
  7:  508,  2850,  4293,  2376,   405;
  8: 1159,  7284, 12717,  8640,  2025;
  9: 2683, 18882, 38421, 32940, 12150, 1458;
%F Formula
The row polynomials satisfy (26*n+26+(13*n^2+45*n+32)*x+(3*n^3+12*n^2+15*n+6)*x^2)*P(n,x) = (182*n+26+(130*n^2+510*n+32)*x+(21*n^3+222*n^2+435*n+6)*x^2+(18*n^3+90*n^2+108*n)*x^3)*P(n-1,x) - (169+(169*n+741)*x+(39*n^2+520*n+1190)*x^2+(75*n^2+531*n+834)*x^3+(36*n^2+180*n+216)*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[((182*n + 26 + (130*n^2 + 510*n + 32)*x + (21*n^3 + 222*n^2 + 435*n + 6)*x^2 + (18*n^3 + 90*n^2 + 108*n)*x^3)*P[n - 1, x] - (169 + (169*n + 741)*x + (39*n^2 + 520*n + 1190)*x^2 + (75*n^2 + 531*n + 834)*x^3 + (36*n^2 + 180*n + 216)*x^4)*D[P[n - 1, x], x])/(26*n + 26 + (13*n^2 + 45*n + 32)*x + (3*n^3 + 12*n^2 + 15*n + 6)*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] = ((182*n + 26 + (130*n^2 + 510*n + 32)*x + (21*n^3 + 222*n^2 + 435*n + 6)*x^2 + (18*n^3 + 90*n^2 + 108*n)*x^3)*v[i - 1] - (169 + (169*n + 741)*x + (39*n^2 + 520*n + 1190)*x^2 + (75*n^2 + 531*n + 834)*x^3 + (36*n^2 + 180*n + 216)*x^4)*deriv(v[i - 1], x))/(26*n + 26 + (13*n^2 + 45*n + 32)*x + (3*n^3 + 12*n^2 + 15*n + 6)*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. [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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.