Real-rooted atlas

A123160 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(126 - 253\,n + 167\,n^{2} - 44\,n^{3} + 4\,n^{4}\right)\,P_{n}(x) = \left(-300 + 455\,n - 239\,n^{2} + 52\,n^{3} - 4\,n^{4}\right)\,P_{n-4}(x) + \left(1056 + 4320\,x - 1640\,n - 6648\,n x + 888\,n^{2} + 3568\,n^{2} x - 200\,n^{3} - 800\,n^{3} x + 16\,n^{4} + 64\,n^{4} x\right)\,P_{n-3}(x) + \left(-4320\,x^{2} + 6648\,n x^{2} - 3568\,n^{2} x^{2} + 800\,n^{3} x^{2} - 64\,n^{4} x^{2}\right)\,P_{n-3}'(x) + \left(-1350 - 7344\,x + 34020\,x^{2} + 2172\,n + 11712\,n x - 53568\,n x^{2} - 1226\,n^{2} - 6560\,n^{2} x + 29664\,n^{2} x^{2} + 288\,n^{3} + 1536\,n^{3} x - 6912\,n^{3} x^{2} - 24\,n^{4} - 128\,n^{4} x + 576\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(7344\,x^{2} + 15120\,x^{3} - 11712\,n x^{2} - 23808\,n x^{3} + 6560\,n^{2} x^{2} + 13184\,n^{2} x^{3} - 1536\,n^{3} x^{2} - 3072\,n^{3} x^{3} + 128\,n^{4} x^{2} + 256\,n^{4} x^{3}\right)\,P_{n-2}'(x) + \left(720 + 3024\,x - 1240\,n - 5064\,n x + 744\,n^{2} + 2992\,n^{2} x - 184\,n^{3} - 736\,n^{3} x + 16\,n^{4} + 64\,n^{4} x\right)\,P_{n-1}(x) + \left(-3024\,x^{2} + 5064\,n x^{2} - 2992\,n^{2} x^{2} + 736\,n^{3} x^{2} - 64\,n^{4} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 4\,x\)
\(P_{3}(x) = 1 + 16\,x + 27\,x^{2}\)
\(P_{4}(x) = 1 + 36\,x + 162\,x^{2} + 160\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2145
31162744
4136162160359
516454012808752760
61100135056008750453620337
71144283517920472505443222638145220
811965292470401837503492723169321098241012307

All 15 rows

Technical details
Node id
n-8b65fbf19b8d0732 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-c3466227e6835c89

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,   4;
   3: 1,  16,    27;
   4: 1,  36,   162,    160;
   5: 1,  64,   540,   1280,     875;
   6: 1, 100,  1350,   5600,    8750,    4536;
   7: 1, 144,  2835,  17920,   47250,   54432,    22638;
   8: 1, 196,  5292,  47040,  183750,  349272,   316932,   109824;
   9: 1, 256,  9072, 107520,  577500, 1596672,  2354352,  1757184,  521235;
  10: 1, 324, 14580, 221760, 1559250, 5837832, 12360348, 14826240, 9382230, 2431000;
%F Formula
The row polynomials satisfy (4*n^4-44*n^3+167*n^2-253*n+126)*P(n,x) = -(4*n^4-52*n^3+239*n^2-455*n+300)*P(n-4,x) + (16*n^4-200*n^3+888*n^2-1640*n+1056+(64*n^4-800*n^3+3568*n^2-6648*n+4320)*x)*P(n-3,x) - (64*n^4-800*n^3+3568*n^2-6648*n+4320)*x^2*d/dx P(n-3,x) - (24*n^4-288*n^3+1226*n^2-2172*n+1350+(128*n^4-1536*n^3+6560*n^2-11712*n+7344)*x-(576*n^4-6912*n^3+29664*n^2-53568*n+34020)*x^2)*P(n-2,x) + ((128*n^4-1536*n^3+6560*n^2-11712*n+7344)*x^2+(256*n^4-3072*n^3+13184*n^2-23808*n+15120)*x^3)*d/dx P(n-2,x) + (16*n^4-184*n^3+744*n^2-1240*n+720+(64*n^4-736*n^3+2992*n^2-5064*n+3024)*x)*P(n-1,x) - (64*n^4-736*n^3+2992*n^2-5064*n+3024)*x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + 4*x, P(3,x) = 1 + 16*x + 27*x^2, P(4,x) = 1 + 36*x + 162*x^2 + 160*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[3, x_] := 1 + 16*x + 27*x^2;
P[4, x_] := 1 + 36*x + 162*x^2 + 160*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*P[n - 4, x] + (16*n^4 - 200*n^3 + 888*n^2 - 1640*n + 1056 + (64*n^4 - 800*n^3 + 3568*n^2 - 6648*n + 4320)*x)*P[n - 3, x] - (64*n^4 - 800*n^3 + 3568*n^2 - 6648*n + 4320)*x^2*D[P[n - 3, x], x] - (24*n^4 - 288*n^3 + 1226*n^2 - 2172*n + 1350 + (128*n^4 - 1536*n^3 + 6560*n^2 - 11712*n + 7344)*x - (576*n^4 - 6912*n^3 + 29664*n^2 - 53568*n + 34020)*x^2)*P[n - 2, x] + ((128*n^4 - 1536*n^3 + 6560*n^2 - 11712*n + 7344)*x^2 + (256*n^4 - 3072*n^3 + 13184*n^2 - 23808*n + 15120)*x^3)*D[P[n - 2, x], x] + (16*n^4 - 184*n^3 + 744*n^2 - 1240*n + 720 + (64*n^4 - 736*n^3 + 2992*n^2 - 5064*n + 3024)*x)*P[n - 1, x] - (64*n^4 - 736*n^3 + 2992*n^2 - 5064*n + 3024)*x^2*D[P[n - 1, x], x])/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)]];
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 + 4*x; v[3] = 1 + 16*x + 27*x^2; v[4] = 1 + 36*x + 162*x^2 + 160*x^3; for(n = 5, N, my(i = n); v[i] = (-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*v[i - 4] + (16*n^4 - 200*n^3 + 888*n^2 - 1640*n + 1056 + (64*n^4 - 800*n^3 + 3568*n^2 - 6648*n + 4320)*x)*v[i - 3] - (64*n^4 - 800*n^3 + 3568*n^2 - 6648*n + 4320)*x^2*deriv(v[i - 3], x) - (24*n^4 - 288*n^3 + 1226*n^2 - 2172*n + 1350 + (128*n^4 - 1536*n^3 + 6560*n^2 - 11712*n + 7344)*x - (576*n^4 - 6912*n^3 + 29664*n^2 - 53568*n + 34020)*x^2)*v[i - 2] + ((128*n^4 - 1536*n^3 + 6560*n^2 - 11712*n + 7344)*x^2 + (256*n^4 - 3072*n^3 + 13184*n^2 - 23808*n + 15120)*x^3)*deriv(v[i - 2], x) + (16*n^4 - 184*n^3 + 744*n^2 - 1240*n + 720 + (64*n^4 - 736*n^3 + 2992*n^2 - 5064*n + 3024)*x)*v[i - 1] - (64*n^4 - 736*n^3 + 2992*n^2 - 5064*n + 3024)*x^2*deriv(v[i - 1], x))/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.