Real-rooted atlas

A128099 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6\,n + 13\,n x + 9\,n^{2} x + 6\,n^{2} x^{2} + 6\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-24 - 160\,x - 312\,x^{2} - 144\,x^{3} + 30\,n + 119\,n x + 72\,n x^{2} - 72\,n x^{3} + 63\,n^{2} x + 222\,n^{2} x^{2} + 144\,n^{2} x^{3} + 30\,n^{3} x^{2} + 72\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-9 - 90\,x - 329\,x^{2} - 516\,x^{3} - 288\,x^{4} - 27\,n x - 207\,n x^{2} - 522\,n x^{3} - 432\,n x^{4} - 18\,n^{2} x^{2} - 102\,n^{2} x^{3} - 144\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
231215
352429
41184108203
521192324537
64349212968642695
7851128367234568341
817126281069215552648035523
93415904281885184032400118673
10683132127322416848016200046656464255
111365291121827364976646156002799361606413
122731637324480921420416220320015396483265926004411
1354611383361075572387763271668806531840228614421081865
1410923298572254534410303200222264002547417613716864223948876814967
Technical details
Node id
n-e186aa55537c94a4 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-811593f461e343e6

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,   12;
  3:   5,   24;
  4:  11,   84,   108;
  5:  21,  192,   324;
  6:  43,  492,  1296,   864;
  7:  85, 1128,  3672,  3456;
  8: 171, 2628, 10692, 15552,  6480;
  9: 341, 5904, 28188, 51840, 32400;
%F Formula
The row polynomials satisfy (6*n+(9*n^2+13*n)*x+(6*n^3+6*n^2)*x^2)*P(n,x) = (30*n-24+(63*n^2+119*n-160)*x+(30*n^3+222*n^2+72*n-312)*x^2+(72*n^3+144*n^2-72*n-144)*x^3)*P(n-1,x) - (9+(27*n+90)*x+(18*n^2+207*n+329)*x^2+(102*n^2+522*n+516)*x^3+(144*n^2+432*n+288)*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[((30*n - 24 + (63*n^2 + 119*n - 160)*x + (30*n^3 + 222*n^2 + 72*n - 312)*x^2 + (72*n^3 + 144*n^2 - 72*n - 144)*x^3)*P[n - 1, x] - (9 + (27*n + 90)*x + (18*n^2 + 207*n + 329)*x^2 + (102*n^2 + 522*n + 516)*x^3 + (144*n^2 + 432*n + 288)*x^4)*D[P[n - 1, x], x])/(6*n + (9*n^2 + 13*n)*x + (6*n^3 + 6*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] = ((30*n - 24 + (63*n^2 + 119*n - 160)*x + (30*n^3 + 222*n^2 + 72*n - 312)*x^2 + (72*n^3 + 144*n^2 - 72*n - 144)*x^3)*v[i - 1] - (9 + (27*n + 90)*x + (18*n^2 + 207*n + 329)*x^2 + (102*n^2 + 522*n + 516)*x^3 + (144*n^2 + 432*n + 288)*x^4)*deriv(v[i - 1], x))/(6*n + (9*n^2 + 13*n)*x + (6*n^3 + 6*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. [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.