Real-rooted atlas

A008278 → x ↦ x+1 (shift +1) → D+1

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(\left(1 + 2\,x + n\right)\,P_{n}(x) = \left(2 + 4\,x + 2\,x^{2} + 2\,x^{3} + 2\,x^{4} + n + 3\,n x + 2\,n x^{2} - n x^{3} - 2\,n x^{4} - 2\,n^{2} - 5\,n^{2} x - 6\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2 + 8\,x + 14\,x^{2} + 14\,x^{3} + 8\,x^{4} + 2\,x^{5} + 3\,n + 11\,n x + 16\,n x^{2} + 11\,n x^{3} + 3\,n x^{4} + n^{2} + 3\,n^{2} x + 3\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(1 + 2\,x^{3} + 3\,n + 6\,n x + 5\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-1 - 4\,x - 5\,x^{2} - 2\,x^{3} - n - 2\,n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
2107118
3384313195
4161261133231579
5747160612033684113966
637531015810253472897675129944
7202536652985475539731721625531411246141

Show all 15 stored rows

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Technical details
Node id
n-6aee275bf0e233bb · revision 1 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-39b0f76a8910e04d

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      3,       1;
  2:     10,       7,       1;
  3:     38,      43,      13,       1;
  4:    161,     261,     133,      23,       1;
  5:    747,    1606,    1203,     368,      41,       1;
  6:   3753,   10158,   10253,    4728,     976,      75,      1;
  7:  20253,   66529,   85475,   53973,   17216,    2553,    141,     1;
  8: 116642,  452623,  711457,  578315,  256826,   60359,   6679,   271,   1;
  9: 713130, 3202057, 5981815, 5997680, 3468608, 1154669, 207985, 17614, 529, 1;
%F Formula
The row polynomials satisfy (n+1+2*x)*P(n,x) = -(n^3+2*n^2-n-2+(2*n^3+5*n^2-3*n-4)*x+(n^3+6*n^2-2*n-2)*x^2+(3*n^2+n-2)*x^3+(2*n-2)*x^4)*P(n-2,x) + (n^2+3*n+2+(3*n^2+11*n+8)*x+(3*n^2+16*n+14)*x^2+(n^2+11*n+14)*x^3+(3*n+8)*x^4+2*x^5)*d/dx P(n-2,x) + (2*n^2+3*n+1+(2*n^2+6*n)*x+5*n*x^2+2*x^3)*P(n-1,x) - (n+1+(2*n+4)*x+(n+5)*x^2+2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 2*n^2 - n - 2 + (2*n^3 + 5*n^2 - 3*n - 4)*x + (n^3 + 6*n^2 - 2*n - 2)*x^2 + (3*n^2 + n - 2)*x^3 + (2*n - 2)*x^4)*P[n - 2, x] + (n^2 + 3*n + 2 + (3*n^2 + 11*n + 8)*x + (3*n^2 + 16*n + 14)*x^2 + (n^2 + 11*n + 14)*x^3 + (3*n + 8)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (2*n^2 + 3*n + 1 + (2*n^2 + 6*n)*x + 5*n*x^2 + 2*x^3)*P[n - 1, x] - (n + 1 + (2*n + 4)*x + (n + 5)*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 1 + 2*x)]];
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 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 2*n^2 - n - 2 + (2*n^3 + 5*n^2 - 3*n - 4)*x + (n^3 + 6*n^2 - 2*n - 2)*x^2 + (3*n^2 + n - 2)*x^3 + (2*n - 2)*x^4)*v[i - 2] + (n^2 + 3*n + 2 + (3*n^2 + 11*n + 8)*x + (3*n^2 + 16*n + 14)*x^2 + (n^2 + 11*n + 14)*x^3 + (3*n + 8)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (2*n^2 + 3*n + 1 + (2*n^2 + 6*n)*x + 5*n*x^2 + 2*x^3)*v[i - 1] - (n + 1 + (2*n + 4)*x + (n + 5)*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 - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-5df37d25263414d4
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-5df37d25263414d4
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-5df37d25263414d4

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

No candidate equalities or related sequences.

References

  1. [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
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.