Real-rooted atlas

A371080: Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3.

basis-map coefficients of A371080, shifted → x ↦ x−1 → basis A008277

A371080 nonnegative coefficients

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

Recurrence from OEIS A371080
\(P_{n}(x) = \left(-3 + x + 3\,n\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1011
20415
302812141
40280160241465
50364025205204016721
605824046480118801280601117941
7011065609878402959604004026608412433145

Show all 15 stored rows

Generated b-files: A035469, A049119 (row sums), A371080 ?

Computed in your browser ?

Technical details
Node id
n-706703e52ab62d2a · revision 1 · JSON
Definition source
oeis_catalog:A371080
Fingerprint
f6-c3894e4490ebd69f

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: 0,         1;
  2: 0,         4,         1;
  3: 0,        28,        12,         1;
  4: 0,       280,       160,        24,        1;
  5: 0,      3640,      2520,       520,       40,       1;
  6: 0,     58240,     46480,     11880,     1280,      60,      1;
  7: 0,   1106560,    987840,    295960,    40040,    2660,     84,    1;
  8: 0,  24344320,  23826880,   8090880,  1296960,  109200,   4928,  112,   1;
  9: 0, 608608000, 643843200, 242280640, 44405760, 4463760, 257040, 8400, 144, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3*n-3+x)*P(n-1,x) + x*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[(3*n - 3 + x)*P[n - 1, x] + x*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] = (3*n - 3 + x)*v[i - 1] + x*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.

A035469: triangle, offset 1, from the row P1, low zeros dropped
%e Example
Triangle begins:
   1:           1;
   2:           4,           1;
   3:          28,          12,          1;
   4:         280,         160,         24,          1;
   5:        3640,        2520,        520,         40,         1;
   6:       58240,       46480,      11880,       1280,        60,        1;
   7:     1106560,      987840,     295960,      40040,      2660,       84,      1;
   8:    24344320,    23826880,    8090880,    1296960,    109200,     4928,    112,     1;
   9:   608608000,   643843200,  242280640,   44405760,   4463760,   257040,   8400,   144,   1;
  10: 17041024000, 19280060800, 7912262400, 1618859200, 187246080, 12946080, 542640, 13440, 180, 1;
%F Formula
Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = (3*n-3+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(1,x) = x.
%t Mathematica
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[(3*n - 3 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = (3*n - 3 + x)*v[i - 1] + x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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.

A371080: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 0,         1;
  2: 0,         4,         1;
  3: 0,        28,        12,         1;
  4: 0,       280,       160,        24,        1;
  5: 0,      3640,      2520,       520,       40,       1;
  6: 0,     58240,     46480,     11880,     1280,      60,      1;
  7: 0,   1106560,    987840,    295960,    40040,    2660,     84,    1;
  8: 0,  24344320,  23826880,   8090880,  1296960,  109200,   4928,  112,   1;
  9: 0, 608608000, 643843200, 242280640, 44405760, 4463760, 257040, 8400, 144, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3*n-3+x)*P(n-1,x) + x*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[(3*n - 3 + x)*P[n - 1, x] + x*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] = (3*n - 3 + x)*v[i - 1] + x*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 >= 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-9c5e956579680e26
  • 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-9c5e956579680e26
  • 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-9c5e956579680e26

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

Same fingerprint (1)

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