Real-rooted atlas

A304236 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(78\,n + 171\,n x + 117\,n^{2} x + 81\,n^{2} x^{2} + 81\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-468 - 3132\,x - 6156\,x^{2} - 2916\,x^{3} + 546\,n + 2250\,n x + 1458\,n x^{2} - 1458\,n x^{3} + 1170\,n^{2} x + 4293\,n^{2} x^{2} + 2916\,n^{2} x^{3} + 567\,n^{3} x^{2} + 1458\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-169 - 1716\,x - 6381\,x^{2} - 10206\,x^{3} - 5832\,x^{4} - 507\,n x - 3978\,n x^{2} - 10287\,n x^{3} - 8748\,n x^{4} - 351\,n^{2} x^{2} - 2025\,n^{2} x^{3} - 2916\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
241822
373643
419162243424
5403967291165
6971224364529167882
72173132111781166426191
85088550386376415232805144652
9115921852114453233280164025534769
102683566463457898893809841503542942632942
11616014306498415029626564100625212576410322419
1214209361296278623897802641705860013817466372008747538160
1332689901368767005230478032626903556448150826040609192294613
14753162241018209062629325368022389412529016678618228426338263752851085202
Technical details
Node id
n-742b4660db4e0e6b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-4fcd95b1b2b8bdea

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:    4,    18;
  3:    7,    36;
  4:   19,   162,    243;
  5:   40,   396,    729;
  6:   97,  1224,   3645,   2916;
  7:  217,  3132,  11178,  11664;
  8:  508,  8550,  38637,  64152,  32805;
  9: 1159, 21852, 114453, 233280, 164025;
%F Formula
The row polynomials satisfy (78*n+(117*n^2+171*n)*x+(81*n^3+81*n^2)*x^2)*P(n,x) = (546*n-468+(1170*n^2+2250*n-3132)*x+(567*n^3+4293*n^2+1458*n-6156)*x^2+(1458*n^3+2916*n^2-1458*n-2916)*x^3)*P(n-1,x) - (169+(507*n+1716)*x+(351*n^2+3978*n+6381)*x^2+(2025*n^2+10287*n+10206)*x^3+(2916*n^2+8748*n+5832)*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[((546*n - 468 + (1170*n^2 + 2250*n - 3132)*x + (567*n^3 + 4293*n^2 + 1458*n - 6156)*x^2 + (1458*n^3 + 2916*n^2 - 1458*n - 2916)*x^3)*P[n - 1, x] - (169 + (507*n + 1716)*x + (351*n^2 + 3978*n + 6381)*x^2 + (2025*n^2 + 10287*n + 10206)*x^3 + (2916*n^2 + 8748*n + 5832)*x^4)*D[P[n - 1, x], x])/(78*n + (117*n^2 + 171*n)*x + (81*n^3 + 81*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] = ((546*n - 468 + (1170*n^2 + 2250*n - 3132)*x + (567*n^3 + 4293*n^2 + 1458*n - 6156)*x^2 + (1458*n^3 + 2916*n^2 - 1458*n - 2916)*x^3)*v[i - 1] - (169 + (507*n + 1716)*x + (351*n^2 + 3978*n + 6381)*x^2 + (2025*n^2 + 10287*n + 10206)*x^3 + (2916*n^2 + 8748*n + 5832)*x^4)*deriv(v[i - 1], x))/(78*n + (117*n^2 + 171*n)*x + (81*n^3 + 81*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.