Real-rooted atlas

A060187: Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k).

A060187 palindromic γ-positive nonnegative coefficients

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

Recurrence from OEIS A060187
\(P_{n}(x) = \left(1 - 3\,x + 2\,n x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2112
31618
412323148
5176230761384
612371682168223713840
71722105432354810543722146080
812179606572597232597236065721791645120

Show all 15 stored rows

Generated b-files: A000165 (row sums), A060187, A060187 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-1be5eefbe5bc79ec · revision 1 · JSON
Definition source
oeis_catalog:A060187
Fingerprint
f6-57e3a3defd926b1b

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: yes
  • γ-positive (gamma_positive): yes
  • 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,     6,       1;
   4: 1,    23,      23,        1;
   5: 1,    76,     230,       76,        1;
   6: 1,   237,    1682,     1682,      237,        1;
   7: 1,   722,   10543,    23548,    10543,      722,        1;
   8: 1,  2179,   60657,   259723,   259723,    60657,     2179,       1;
   9: 1,  6552,  331612,  2485288,  4675014,  2485288,   331612,    6552,     1;
  10: 1, 19673, 1756340, 21707972, 69413294, 69413294, 21707972, 1756340, 19673, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(2*n-3)*x)*P(n-1,x) + (2*x-2*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 + (2*n - 3)*x)*P[n - 1, x] + (2*x - 2*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 + (2*n - 3)*x)*v[i - 1] + (2*x - 2*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.

A060187: triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1: 1;
   2: 1,     1;
   3: 1,     6,       1;
   4: 1,    23,      23,        1;
   5: 1,    76,     230,       76,        1;
   6: 1,   237,    1682,     1682,      237,        1;
   7: 1,   722,   10543,    23548,    10543,      722,        1;
   8: 1,  2179,   60657,   259723,   259723,    60657,     2179,       1;
   9: 1,  6552,  331612,  2485288,  4675014,  2485288,   331612,    6552,     1;
  10: 1, 19673, 1756340, 21707972, 69413294, 69413294, 21707972, 1756340, 19673, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(2*n-3)*x)*P(n-1,x) + (2*x-2*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 + (2*n - 3)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {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 + (2*n - 3)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A060187: reversed triangle, offset 1, from the row P1
%e Example
Triangle begins:
   1: 1;
   2: 1,     1;
   3: 1,     6,       1;
   4: 1,    23,      23,        1;
   5: 1,    76,     230,       76,        1;
   6: 1,   237,    1682,     1682,      237,        1;
   7: 1,   722,   10543,    23548,    10543,      722,        1;
   8: 1,  2179,   60657,   259723,   259723,    60657,     2179,       1;
   9: 1,  6552,  331612,  2485288,  4675014,  2485288,   331612,    6552,     1;
  10: 1, 19673, 1756340, 21707972, 69413294, 69413294, 21707972, 1756340, 19673, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(2*n-3)*x)*P(n-1,x) + (2*x-2*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 + (2*n - 3)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {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 + (2*n - 3)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 1, 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-974c6818ff00e631
  • 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-974c6818ff00e631
  • 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-974c6818ff00e631
  • 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-974c6818ff00e631

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 3 proof steps

Same fingerprint (3)

Equal up to content, sign, reversal and a small index shift.

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