Real-rooted atlas

A008292 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 2\,x + n\right)\,P_{n}(x) = \left(-4\,x - 10\,x^{2} - 6\,x^{3} + 4\,x^{4} - 2\,n + n x + 11\,n x^{2} + 8\,n x^{3} - 2\,n x^{4} + 3\,n^{2} + 5\,n^{2} x - n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x^{2} - 4\,x^{3} + 2\,x^{5} - n x + 4\,n x^{3} + 3\,n x^{4} + n^{2} x + 2\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(1 - x - 8\,x^{2} + 2\,x^{3} - 3\,n - n x + 5\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(x - x^{2} - 2\,x^{3} - n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2314
3128121
46064171142
53605402403411175
625204920318078867111476
72016048720420001562424361321129073
818144052416056952028929669804732026111641802

All 15 rows

Technical details
Node id
n-65c63cabff9a971a · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-08aabc8d9547846a

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:        3,        1;
   3:       12,        8,         1;
   4:       60,       64,        17,        1;
   5:      360,      540,       240,       34,        1;
   6:     2520,     4920,      3180,      788,       67,        1;
   7:    20160,    48720,     42000,    15624,     2436,      132,       1;
   8:   181440,   524160,    569520,   289296,    69804,     7320,     261,     1;
   9:  1814400,  6108480,   8043840,  5241600,  1766520,   295380,   21720,   518,    1;
  10: 19958400, 76809600, 119145600, 95376960, 41950440, 10014120, 1210380, 64156, 1031, 1;
%F Formula
The row polynomials satisfy (n-1+2*x)*P(n,x) = -(n^3-3*n^2+2*n+(2*n^3-5*n^2-n+4)*x+(n^3+n^2-11*n+10)*x^2+(3*n^2-8*n+6)*x^3+(2*n-4)*x^4)*P(n-2,x) + ((n^2-n)*x+(2*n^2-2)*x^2+(n^2+4*n-4)*x^3+3*n*x^4+2*x^5)*d/dx P(n-2,x) + (2*n^2-3*n+1+(2*n^2-n-1)*x+(5*n-8)*x^2+2*x^3)*P(n-1,x) - ((n-1)*x+(n+1)*x^2+2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 3*n^2 + 2*n + (2*n^3 - 5*n^2 - n + 4)*x + (n^3 + n^2 - 11*n + 10)*x^2 + (3*n^2 - 8*n + 6)*x^3 + (2*n - 4)*x^4)*P[n - 2, x] + ((n^2 - n)*x + (2*n^2 - 2)*x^2 + (n^2 + 4*n - 4)*x^3 + 3*n*x^4 + 2*x^5)*D[P[n - 2, x], x] + (2*n^2 - 3*n + 1 + (2*n^2 - n - 1)*x + (5*n - 8)*x^2 + 2*x^3)*P[n - 1, x] - ((n - 1)*x + (n + 1)*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n - 1 + 2*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] = 3 + x; for(n = 3, N, my(i = n); v[i] = (-(n^3 - 3*n^2 + 2*n + (2*n^3 - 5*n^2 - n + 4)*x + (n^3 + n^2 - 11*n + 10)*x^2 + (3*n^2 - 8*n + 6)*x^3 + (2*n - 4)*x^4)*v[i - 2] + ((n^2 - n)*x + (2*n^2 - 2)*x^2 + (n^2 + 4*n - 4)*x^3 + 3*n*x^4 + 2*x^5)*deriv(v[i - 2], x) + (2*n^2 - 3*n + 1 + (2*n^2 - n - 1)*x + (5*n - 8)*x^2 + 2*x^3)*v[i - 1] - ((n - 1)*x + (n + 1)*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n - 1 + 2*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

Same fingerprint ?

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.