Real-rooted atlas

A113214 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 210\,x + 324\,x^{2} + 270\,x^{3} - 81\,x^{4} + 225\,n x - 900\,n x^{2} - 513\,n x^{3} + 162\,n x^{4} + 540\,n^{2} x^{2} - 594\,n^{2} x^{3} - 891\,n^{2} x^{4} + 405\,n^{3} x^{3} - 243\,n^{3} x^{5} + 81\,n^{4} x^{4}\right)\,P_{n}(x) = \left(25 + 90\,x + 9\,x^{2} - 540\,x^{3} - 2025\,x^{4} - 2430\,x^{5} - 729\,x^{6} + 225\,n x + 855\,n x^{2} + 54\,n x^{3} - 2754\,n x^{4} - 4617\,n x^{5} - 2187\,n x^{6} + 540\,n^{2} x^{2} + 2241\,n^{2} x^{3} + 1458\,n^{2} x^{4} - 1944\,n^{2} x^{5} - 2187\,n^{2} x^{6} + 405\,n^{3} x^{3} + 1539\,n^{3} x^{4} + 729\,n^{3} x^{5} - 729\,n^{3} x^{6} + 81\,n^{4} x^{4} + 243\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(25 + 90\,x - 306\,x^{2} - 2106\,x^{3} - 3807\,x^{4} - 1944\,x^{5} + 225\,n x + 540\,n x^{2} - 1917\,n x^{3} - 6075\,n x^{4} - 4860\,n x^{5} - 729\,n x^{6} + 540\,n^{2} x^{2} + 1836\,n^{2} x^{3} - 243\,n^{2} x^{4} - 3159\,n^{2} x^{5} - 1458\,n^{2} x^{6} + 405\,n^{3} x^{3} + 1377\,n^{3} x^{4} + 243\,n^{3} x^{5} - 729\,n^{3} x^{6} + 81\,n^{4} x^{4} + 243\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131215
246081145
371565674321162
4113482106388820258378
51872060751857622275874856412
6291416153906609612960011372435721361976
7472700358831991525487757785725358151399682240912

All 15 rows

Technical details
Node id
n-462f9ddd4c2300fe · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-2f630d379786a4d6

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 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:   3,   12;
  2:   4,   60,     81;
  3:   7,  156,    567,     432;
  4:  11,  348,   2106,    3888,    2025;
  5:  18,  720,   6075,   18576,   22275,     8748;
  6:  29, 1416,  15390,   66096,  129600,   113724,    35721;
  7:  47, 2700,  35883,  199152,  548775,   778572,   535815,   139968;
  8:  76, 5028,  79056,  538704, 1921725,  3822876,  4215078,  2379456,   531441;
  9: 123, 9204, 167103, 1350864, 5929200, 15274008, 23540139, 21135168, 10097379, 1968300;
%F Formula
The row polynomials satisfy (25+(225*n-210)*x+(540*n^2-900*n+324)*x^2+(405*n^3-594*n^2-513*n+270)*x^3+(81*n^4-891*n^2+162*n-81)*x^4-243*n^3*x^5)*P(n,x) = (25+(225*n+90)*x+(540*n^2+855*n+9)*x^2+(405*n^3+2241*n^2+54*n-540)*x^3+(81*n^4+1539*n^3+1458*n^2-2754*n-2025)*x^4+(243*n^4+729*n^3-1944*n^2-4617*n-2430)*x^5-(729*n^3+2187*n^2+2187*n+729)*x^6)*P(n-2,x) + (25+(225*n+90)*x+(540*n^2+540*n-306)*x^2+(405*n^3+1836*n^2-1917*n-2106)*x^3+(81*n^4+1377*n^3-243*n^2-6075*n-3807)*x^4+(243*n^4+243*n^3-3159*n^2-4860*n-1944)*x^5-(729*n^3+1458*n^2+729*n)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[((25 + (225*n + 90)*x + (540*n^2 + 855*n + 9)*x^2 + (405*n^3 + 2241*n^2 + 54*n - 540)*x^3 + (81*n^4 + 1539*n^3 + 1458*n^2 - 2754*n - 2025)*x^4 + (243*n^4 + 729*n^3 - 1944*n^2 - 4617*n - 2430)*x^5 - (729*n^3 + 2187*n^2 + 2187*n + 729)*x^6)*P[n - 2, x] + (25 + (225*n + 90)*x + (540*n^2 + 540*n - 306)*x^2 + (405*n^3 + 1836*n^2 - 1917*n - 2106)*x^3 + (81*n^4 + 1377*n^3 - 243*n^2 - 6075*n - 3807)*x^4 + (243*n^4 + 243*n^3 - 3159*n^2 - 4860*n - 1944)*x^5 - (729*n^3 + 1458*n^2 + 729*n)*x^6)*P[n - 1, x])/(25 + (225*n - 210)*x + (540*n^2 - 900*n + 324)*x^2 + (405*n^3 - 594*n^2 - 513*n + 270)*x^3 + (81*n^4 - 891*n^2 + 162*n - 81)*x^4 - 243*n^3*x^5)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = ((25 + (225*n + 90)*x + (540*n^2 + 855*n + 9)*x^2 + (405*n^3 + 2241*n^2 + 54*n - 540)*x^3 + (81*n^4 + 1539*n^3 + 1458*n^2 - 2754*n - 2025)*x^4 + (243*n^4 + 729*n^3 - 1944*n^2 - 4617*n - 2430)*x^5 - (729*n^3 + 2187*n^2 + 2187*n + 729)*x^6)*v[i - 2] + (25 + (225*n + 90)*x + (540*n^2 + 540*n - 306)*x^2 + (405*n^3 + 1836*n^2 - 1917*n - 2106)*x^3 + (81*n^4 + 1377*n^3 - 243*n^2 - 6075*n - 3807)*x^4 + (243*n^4 + 243*n^3 - 3159*n^2 - 4860*n - 1944)*x^5 - (729*n^3 + 1458*n^2 + 729*n)*x^6)*v[i - 1])/(25 + (225*n - 210)*x + (540*n^2 - 900*n + 324)*x^2 + (405*n^3 - 594*n^2 - 513*n + 270)*x^3 + (81*n^4 - 891*n^2 + 162*n - 81)*x^4 - 243*n^3*x^5)); 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.