Real-rooted atlas

A059364: Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.

A059364 nonnegative coefficients only rational roots

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

Recurrence from OEIS A059364
\(P_{n}(x) = \left(1 + n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
267215
32446296105
412032632914624945
572025563604252187412010395
65040222124056439271212446084720135135
7403202129764799965981164448491973804834850402027025

Show all 15 stored rows

Generated b-files: A059364 ?

Computed in your browser ?

Technical details
Node id
n-7fd16a32c438d522 · revision 3 · JSON
Definition source
oeis_catalog:A059364
Fingerprint
f6-774e45ab36bcae1f

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): yes

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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       2,        1;
  2:       6,        7,        2;
  3:      24,       46,       29,         6;
  4:     120,      326,      329,       146,        24;
  5:     720,     2556,     3604,      2521,       874,       120;
  6:    5040,    22212,    40564,     39271,     21244,      6084,      720;
  7:   40320,   212976,   479996,    598116,    444849,    197380,    48348,     5040;
  8:  362880,  2239344,  6023772,   9223012,   8788569,   5335212,  2014172,   432144,   40320;
  9: 3628800, 25659360, 80391816, 146444068, 170892798, 132449241, 68158628, 22448988, 4292496, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (n+1+n*x)*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[(n + 1 + n*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {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] = (n + 1 + n*x)*v[i - 1]); 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.

A059364: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:       1;
   2:       2,        1;
   3:       6,        7,        2;
   4:      24,       46,       29,         6;
   5:     120,      326,      329,       146,        24;
   6:     720,     2556,     3604,      2521,       874,       120;
   7:    5040,    22212,    40564,     39271,     21244,      6084,      720;
   8:   40320,   212976,   479996,    598116,    444849,    197380,    48348,     5040;
   9:  362880,  2239344,  6023772,   9223012,   8788569,   5335212,  2014172,   432144,   40320;
  10: 3628800, 25659360, 80391816, 146444068, 170892798, 132449241, 68158628, 22448988, 4292496, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (n+(n-1)*x)*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[(n + (n - 1)*x)*P[n - 1, 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] = (n + (n - 1)*x)*v[i - 1]); 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.

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-50fc3242d79d4e9a
  • 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-50fc3242d79d4e9a, e-8e142d34c9c1ef70, e-b8821d58f365ef74, e-bf95690978550a83
  • 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-50fc3242d79d4e9a, e-8e142d34c9c1ef70, e-b8821d58f365ef74, e-bf95690978550a83
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-50fc3242d79d4e9a
  • 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-b8821d58f365ef74, e-bf95690978550a83
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-8e142d34c9c1ef70, e-b8821d58f365ef74
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-8e142d34c9c1ef70

Proof chains

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

Used in 8 proof steps

Same fingerprint (2)

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