Real-rooted atlas

A008277 → θ+1 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 + 20\,x + 190\,x^{2} - 4\,n x - 83\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n-4}(x) + \left(-16\,x^{2} + 140\,x^{3} + 4\,n x^{2} - 50\,n x^{3} + 4\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(17\,x^{4} - 4\,n x^{4}\right)\,P_{n-4}''(x) + \left(4 - 44\,x - 122\,x^{2} + 12\,n x + 103\,n x^{2} - 18\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(32\,x^{2} - 113\,x^{3} - 12\,n x^{2} + 72\,n x^{3} - 8\,n^{2} x^{3}\right)\,P_{n-3}'(x) + \left(-18\,x^{4} + 8\,n x^{4}\right)\,P_{n-3}''(x) + \left(-6 + 28\,x + 4\,x^{2} - 12\,n x - 20\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-16\,x^{2} + 5\,x^{3} + 12\,n x^{2} - 22\,n x^{3} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(x^{4} - 4\,n x^{4}\right)\,P_{n-2}''(x) + \left(4 - 4\,x + 4\,n x\right)\,P_{n-1}(x) + \left(-4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 4\,x\)
\(P_{3}(x) = 3 + 24\,x + 9\,x^{2}\)
\(P_{4}(x) = 4 + 72\,x + 126\,x^{2} + 16\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2246
3324936
447212616218
55160675480251345
66300234043201550368552
775046300224002257545364956371
8878414364840001701001043281244664386094
99115229106254016868875111888044467532640812749434
101016205400065856034240507654500668458017913608278210020351562
11112200935551520640111977253876919260448850373120006925743204600121156474637
12122904153450320760031805400158103792389153688441408000198126324259578004953741441248414488
13133744240669629200080887950547088256196039043235793838083040942905101301200094969875117936016910324391181
1414473236363611627616188188000166378212081976294402209951744030830077278200375175005029604580340835040276855819688401916150
15155880532350204422404068564504558554000296036640901109134540802317563358112524159638001275139866002439702720012041762076422136225782797421084
Technical details
Node id
n-fc94df5213f0c4e4 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b50e0bffdfcf6785

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:  2,    4;
   3:  3,   24,     9;
   4:  4,   72,   126,     16;
   5:  5,  160,   675,    480,      25;
   6:  6,  300,  2340,   4320,    1550,      36;
   7:  7,  504,  6300,  22400,   22575,    4536,      49;
   8:  8,  784, 14364,  84000,  170100,  104328,   12446,      64;
   9:  9, 1152, 29106, 254016,  868875, 1118880,  444675,   32640,    81;
  10: 10, 1620, 54000, 658560, 3424050, 7654500, 6684580, 1791360, 82782, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(1+(4*n-20)*x-(9*n^2-83*n+190)*x^2)*P(n-4,x) + ((4*n-16)*x^2+(4*n^2-50*n+140)*x^3)*d/dx P(n-4,x) - (4*n-17)*x^4*d^2/dx^2 P(n-4,x) + (4+(12*n-44)*x-(18*n^2-103*n+122)*x^2)*P(n-3,x) - ((12*n-32)*x^2+(8*n^2-72*n+113)*x^3)*d/dx P(n-3,x) + (8*n-18)*x^4*d^2/dx^2 P(n-3,x) - (6+(12*n-28)*x-(9*n^2-20*n+4)*x^2)*P(n-2,x) + ((12*n-16)*x^2+(4*n^2-22*n+5)*x^3)*d/dx P(n-2,x) - (4*n-1)*x^4*d^2/dx^2 P(n-2,x) + (4+(4*n-4)*x)*P(n-1,x) - 4*n*x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2 + 4*x, P(3,x) = 3 + 24*x + 9*x^2, P(4,x) = 4 + 72*x + 126*x^2 + 16*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 4*x;
P[3, x_] := 3 + 24*x + 9*x^2;
P[4, x_] := 4 + 72*x + 126*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(1 + (4*n - 20)*x - (9*n^2 - 83*n + 190)*x^2)*P[n - 4, x] + ((4*n - 16)*x^2 + (4*n^2 - 50*n + 140)*x^3)*D[P[n - 4, x], x] - (4*n - 17)*x^4*D[P[n - 4, x], {x, 2}] + (4 + (12*n - 44)*x - (18*n^2 - 103*n + 122)*x^2)*P[n - 3, x] - ((12*n - 32)*x^2 + (8*n^2 - 72*n + 113)*x^3)*D[P[n - 3, x], x] + (8*n - 18)*x^4*D[P[n - 3, x], {x, 2}] - (6 + (12*n - 28)*x - (9*n^2 - 20*n + 4)*x^2)*P[n - 2, x] + ((12*n - 16)*x^2 + (4*n^2 - 22*n + 5)*x^3)*D[P[n - 2, x], x] - (4*n - 1)*x^4*D[P[n - 2, x], {x, 2}] + (4 + (4*n - 4)*x)*P[n - 1, x] - 4*n*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; v[3] = 3 + 24*x + 9*x^2; v[4] = 4 + 72*x + 126*x^2 + 16*x^3; for(n = 5, N, my(i = n); v[i] = -(1 + (4*n - 20)*x - (9*n^2 - 83*n + 190)*x^2)*v[i - 4] + ((4*n - 16)*x^2 + (4*n^2 - 50*n + 140)*x^3)*deriv(v[i - 4], x) - (4*n - 17)*x^4*deriv(deriv(v[i - 4], x), x) + (4 + (12*n - 44)*x - (18*n^2 - 103*n + 122)*x^2)*v[i - 3] - ((12*n - 32)*x^2 + (8*n^2 - 72*n + 113)*x^3)*deriv(v[i - 3], x) + (8*n - 18)*x^4*deriv(deriv(v[i - 3], x), x) - (6 + (12*n - 28)*x - (9*n^2 - 20*n + 4)*x^2)*v[i - 2] + ((12*n - 16)*x^2 + (4*n^2 - 22*n + 5)*x^3)*deriv(v[i - 2], x) - (4*n - 1)*x^4*deriv(deriv(v[i - 2], x), x) + (4 + (4*n - 4)*x)*v[i - 1] - 4*n*x^2*deriv(v[i - 1], x)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.