Real-rooted atlas

A102537 → D+1

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 4\,n x - 2\,n x^{2} + 10\,n^{2} x - 4\,n^{2} x^{2} - 12\,n^{3} + 4\,n^{3} x + 8\,n^{4}\right)\,P_{n}(x) = \left(4\,n x - 8\,n x^{2} + 6\,n x^{3} + 16\,n^{2} x + 8\,n^{2} x^{2} - 18\,n^{2} x^{3} + 4\,n^{3} - 26\,n^{3} x + 18\,n^{3} x^{2} + 8\,n^{4} + 36\,n^{4} x\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2} + 2\,x^{3} + 3\,x^{4} + 10\,n x + 12\,n x^{2} - 11\,n x^{3} - 9\,n x^{4} - 18\,n^{2} x - 16\,n^{2} x^{2} + 9\,n^{2} x^{3} + 16\,n^{3} x + 18\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2437
39321253
41612522055416
525336124814562733338
6367354550109209520142827189
7491408129205059289148620167752223885
864245731122177555502740697851403788432631858840

All 15 rows

Technical details
Node id
n-69c049f5da53d5d7 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a919b4b97d298de5

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:   4,    3;
   3:   9,   32,     12;
   4:  16,  125,    220,      55;
   5:  25,  336,   1248,    1456,     273;
   6:  36,  735,   4550,   10920,    9520,     1428;
   7:  49, 1408,  12920,   50592,   89148,    62016,     7752;
   8:  64, 2457,  31122,  177555,  502740,   697851,   403788,    43263;
   9:  81, 4000,  66640,  517440, 2107490,  4647104,  5313000,  2631200,   246675;
  10: 100, 6171, 130548, 1317624, 7225680, 22694100, 40849380, 39665340, 17168580, 1430715;
%F Formula
The row polynomials satisfy (8*n^4-12*n^3+4*n+(4*n^3+10*n^2+4*n)*x-(4*n^2+2*n)*x^2)*P(n,x) = (8*n^4+4*n^3+(36*n^4-26*n^3+16*n^2+4*n)*x+(18*n^3+8*n^2-8*n)*x^2-(18*n^2-6*n)*x^3)*P(n-1,x) + ((16*n^3-18*n^2+10*n-2)*x+(18*n^3-16*n^2+12*n-2)*x^2+(9*n^2-11*n+2)*x^3-(9*n-3)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^4 + 4*n^3 + (36*n^4 - 26*n^3 + 16*n^2 + 4*n)*x + (18*n^3 + 8*n^2 - 8*n)*x^2 - (18*n^2 - 6*n)*x^3)*P[n - 1, x] + ((16*n^3 - 18*n^2 + 10*n - 2)*x + (18*n^3 - 16*n^2 + 12*n - 2)*x^2 + (9*n^2 - 11*n + 2)*x^3 - (9*n - 3)*x^4)*D[P[n - 1, x], x])/(8*n^4 - 12*n^3 + 4*n + (4*n^3 + 10*n^2 + 4*n)*x - (4*n^2 + 2*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^4 + 4*n^3 + (36*n^4 - 26*n^3 + 16*n^2 + 4*n)*x + (18*n^3 + 8*n^2 - 8*n)*x^2 - (18*n^2 - 6*n)*x^3)*v[i - 1] + ((16*n^3 - 18*n^2 + 10*n - 2)*x + (18*n^3 - 16*n^2 + 12*n - 2)*x^2 + (9*n^2 - 11*n + 2)*x^3 - (9*n - 3)*x^4)*deriv(v[i - 1], x))/(8*n^4 - 12*n^3 + 4*n + (4*n^3 + 10*n^2 + 4*n)*x - (4*n^2 + 2*n)*x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.