Real-rooted atlas

A026729 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 255\,x + 306\,x^{2} + 702\,x^{3} + 81\,x^{4} - 243\,x^{5} + 225\,n x - 990\,n x^{2} - 486\,n x^{3} + 1134\,n x^{4} + 243\,n x^{5} + 540\,n^{2} x^{2} - 675\,n^{2} x^{3} - 1053\,n^{2} x^{4} + 243\,n^{2} x^{5} + 405\,n^{3} x^{3} - 243\,n^{3} x^{5} + 81\,n^{4} x^{4}\right)\,P_{n}(x) = \left(25 + 45\,x - 234\,x^{2} - 486\,x^{3} + 81\,x^{4} + 729\,x^{5} + 225\,n x + 765\,n x^{2} - 351\,n x^{3} - 2511\,n x^{4} - 1944\,n x^{5} + 540\,n^{2} x^{2} + 2160\,n^{2} x^{3} + 1053\,n^{2} x^{4} - 2187\,n^{2} x^{5} - 1458\,n^{2} x^{6} + 405\,n^{3} x^{3} + 1539\,n^{3} x^{4} + 729\,n^{3} x^{5} - 729\,n^{3} x^{6} + 81\,n^{4} x^{4} + 243\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(25 + 45\,x - 234\,x^{2} - 486\,x^{3} + 81\,x^{4} + 729\,x^{5} + 225\,n x + 450\,n x^{2} - 1836\,n x^{3} - 3888\,n x^{4} - 1215\,n x^{5} + 540\,n^{2} x^{2} + 1755\,n^{2} x^{3} - 486\,n^{2} x^{4} - 2916\,n^{2} x^{5} - 729\,n^{2} x^{6} + 405\,n^{3} x^{3} + 1377\,n^{3} x^{4} + 243\,n^{3} x^{5} - 729\,n^{3} x^{6} + 81\,n^{4} x^{4} + 243\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
211213
323681119
4384405432924
551801296302420256530
6836034831252818225874843352
713696842441040931509622835721275272
8211308190351170723624755861164643731399681690368

All 15 rows

Technical details
Node id
n-8bc628d590897076 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-84e29d07816d0d9d

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  1,   12;
   3:  2,   36,    81;
   4:  3,   84,   405,    432;
   5:  5,  180,  1296,   3024,    2025;
   6:  8,  360,  3483,  12528,   18225,    8748;
   7: 13,  696,  8424,  41040,   93150,   96228,    35721;
   8: 21, 1308, 19035, 117072,  362475,  586116,   464373,   139968;
   9: 34, 2412, 40986, 304560, 1196775, 2650644,  3286332,  2099520,  531441;
  10: 55, 4380, 85131, 741744, 3535650, 9972720, 16967475, 16936128, 9034497, 1968300;
%F Formula
The row polynomials satisfy (25+(225*n-255)*x+(540*n^2-990*n+306)*x^2+(405*n^3-675*n^2-486*n+702)*x^3+(81*n^4-1053*n^2+1134*n+81)*x^4-(243*n^3-243*n^2-243*n+243)*x^5)*P(n,x) = (25+(225*n+45)*x+(540*n^2+765*n-234)*x^2+(405*n^3+2160*n^2-351*n-486)*x^3+(81*n^4+1539*n^3+1053*n^2-2511*n+81)*x^4+(243*n^4+729*n^3-2187*n^2-1944*n+729)*x^5-(729*n^3+1458*n^2)*x^6)*P(n-2,x) + (25+(225*n+45)*x+(540*n^2+450*n-234)*x^2+(405*n^3+1755*n^2-1836*n-486)*x^3+(81*n^4+1377*n^3-486*n^2-3888*n+81)*x^4+(243*n^4+243*n^3-2916*n^2-1215*n+729)*x^5-(729*n^3+729*n^2)*x^6)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 12*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[((25 + (225*n + 45)*x + (540*n^2 + 765*n - 234)*x^2 + (405*n^3 + 2160*n^2 - 351*n - 486)*x^3 + (81*n^4 + 1539*n^3 + 1053*n^2 - 2511*n + 81)*x^4 + (243*n^4 + 729*n^3 - 2187*n^2 - 1944*n + 729)*x^5 - (729*n^3 + 1458*n^2)*x^6)*P[n - 2, x] + (25 + (225*n + 45)*x + (540*n^2 + 450*n - 234)*x^2 + (405*n^3 + 1755*n^2 - 1836*n - 486)*x^3 + (81*n^4 + 1377*n^3 - 486*n^2 - 3888*n + 81)*x^4 + (243*n^4 + 243*n^3 - 2916*n^2 - 1215*n + 729)*x^5 - (729*n^3 + 729*n^2)*x^6)*P[n - 1, x])/(25 + (225*n - 255)*x + (540*n^2 - 990*n + 306)*x^2 + (405*n^3 - 675*n^2 - 486*n + 702)*x^3 + (81*n^4 - 1053*n^2 + 1134*n + 81)*x^4 - (243*n^3 - 243*n^2 - 243*n + 243)*x^5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 12*x; for(n = 3, N, my(i = n); v[i] = ((25 + (225*n + 45)*x + (540*n^2 + 765*n - 234)*x^2 + (405*n^3 + 2160*n^2 - 351*n - 486)*x^3 + (81*n^4 + 1539*n^3 + 1053*n^2 - 2511*n + 81)*x^4 + (243*n^4 + 729*n^3 - 2187*n^2 - 1944*n + 729)*x^5 - (729*n^3 + 1458*n^2)*x^6)*v[i - 2] + (25 + (225*n + 45)*x + (540*n^2 + 450*n - 234)*x^2 + (405*n^3 + 1755*n^2 - 1836*n - 486)*x^3 + (81*n^4 + 1377*n^3 - 486*n^2 - 3888*n + 81)*x^4 + (243*n^4 + 243*n^3 - 2916*n^2 - 1215*n + 729)*x^5 - (729*n^3 + 729*n^2)*x^6)*v[i - 1])/(25 + (225*n - 255)*x + (540*n^2 - 990*n + 306)*x^2 + (405*n^3 - 675*n^2 - 486*n + 702)*x^3 + (81*n^4 - 1053*n^2 + 1134*n + 81)*x^4 - (243*n^3 - 243*n^2 - 243*n + 243)*x^5)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.