Real-rooted atlas

A112493: Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and 0 <= k <= n.

A112493 nonnegative coefficients

Proved for all n: all roots real, in [−1, 0) · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence from OEIS A112493
\(P_{n}(x) = \left(1 - x + n x\right)\,P_{n-1}(x) + \left(x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2112
31438
4111251552
5126130210105472
6157546175022059455504
7112020371136826775277201039578416
812477071638052475554608454054051351351320064

Show all 15 stored rows

Generated b-files: A006351 (row sums), A112493 ?

Computed in your browser ?

Technical details
Node id
n-5b857304d1bafe7a · revision 1 · JSON
Definition source
oeis_catalog:A112493
Fingerprint
f6-e733eac92d45f846

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

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,    1;
   3: 1,    4,     3;
   4: 1,   11,    25,      15;
   5: 1,   26,   130,     210,      105;
   6: 1,   57,   546,    1750,     2205,      945;
   7: 1,  120,  2037,   11368,    26775,    27720,     10395;
   8: 1,  247,  7071,   63805,   247555,   460845,    405405,    135135;
   9: 1,  502, 23436,  325930,  1939630,  5735730,   8828820,   6756750,   2027025;
  10: 1, 1013, 75328, 1561516, 13609310, 59629570, 142101960, 186486300, 126351225, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n-1)*x)*P(n-1,x) + (x+x^2)*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[(1 + (n - 1)*x)*P[n - 1, x] + (x + 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; for(n = 2, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 1] + (x + 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)

Generated with polytool c48629374597.

A112493: triangle, offset 0, from the row P1
%e Example
Triangle begins:
  0: 1;
  1: 1,    1;
  2: 1,    4,     3;
  3: 1,   11,    25,      15;
  4: 1,   26,   130,     210,      105;
  5: 1,   57,   546,    1750,     2205,      945;
  6: 1,  120,  2037,   11368,    26775,    27720,     10395;
  7: 1,  247,  7071,   63805,   247555,   460845,    405405,    135135;
  8: 1,  502, 23436,  325930,  1939630,  5735730,   8828820,   6756750,   2027025;
  9: 1, 1013, 75328, 1561516, 13609310, 59629570, 142101960, 186486300, 126351225, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = (1+n*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + n*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + n*x)*v[i - 1] + (x + x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 >= 1, is zero or real-rooted with all roots in [-1, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-6e3ca58296ce0220
  • Coefficients of one sign. Every nonzero row, n >= 1: 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-6e3ca58296ce0220
  • Hadamard kernel (Malo). Rows P_n, n >= 1, 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-6e3ca58296ce0220
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-6e3ca58296ce0220

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

Used in 2 proof steps

No candidate equalities or related sequences.

References

  1. [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