Real-rooted atlas

A028246: Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.

A028246 nonnegative coefficients

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

Recurrence from OEIS A028246
\(P_{n}(x) = \left(1 + x\right)\,P_{n-1}(x) + \left(x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
21326
31712626
4115506024150
51311803903601201082
61636022100336025207209366
71127193210206252003192020160504094586

Show all 15 stored rows

Generated b-files: A000629 (row sums), A028246, A130850 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-677d34c7af372519 · revision 1 · JSON
Definition source
oeis_catalog:A028246
Fingerprint
f6-a33ed99ce72b6651

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 ?

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: 1,   1;
  2: 1,   3,     2;
  3: 1,   7,    12,      6;
  4: 1,  15,    50,     60,      24;
  5: 1,  31,   180,    390,     360,     120;
  6: 1,  63,   602,   2100,    3360,    2520,     720;
  7: 1, 127,  1932,  10206,   25200,   31920,   20160,    5040;
  8: 1, 255,  6050,  46620,  166824,  317520,  332640,  181440,   40320;
  9: 1, 511, 18660, 204630, 1020600, 2739240, 4233600, 3780000, 1814400, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (1+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 + x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], 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] = (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 - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A028246: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,   1;
   3: 1,   3,     2;
   4: 1,   7,    12,      6;
   5: 1,  15,    50,     60,      24;
   6: 1,  31,   180,    390,     360,     120;
   7: 1,  63,   602,   2100,    3360,    2520,     720;
   8: 1, 127,  1932,  10206,   25200,   31920,   20160,    5040;
   9: 1, 255,  6050,  46620,  166824,  317520,  332640,  181440,   40320;
  10: 1, 511, 18660, 204630, 1020600, 2739240, 4233600, 3780000, 1814400, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (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 + x)*P[n - 1, x] + (x + 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 + 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 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A130850: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:      1;
  1:      1,       1;
  2:      2,       3,       1;
  3:      6,      12,       7,       1;
  4:     24,      60,      50,      15,       1;
  5:    120,     360,     390,     180,      31,       1;
  6:    720,    2520,    3360,    2100,     602,      63,      1;
  7:   5040,   20160,   31920,   25200,   10206,    1932,    127,     1;
  8:  40320,  181440,  332640,  317520,  166824,   46620,   6050,   255,   1;
  9: 362880, 1814400, 3780000, 4233600, 2739240, 1020600, 204630, 18660, 511, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+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 + x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];
Table[Reverse[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 + 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 = Vecrev(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 >= 0, 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-994b5c04cf13d0cc
  • 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-994b5c04cf13d0cc
  • 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-994b5c04cf13d0cc
  • 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-994b5c04cf13d0cc
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in [0, 1] to roots in [-1, 0]: source [0, 1], target [-1, 0]
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-4720724f7f8010a1

Proof chains

2 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 3 proof steps

Same fingerprint (2)

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

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