Real-rooted atlas

A163932: Triangle related to the asymptotic expansion of E(x,m=3,n).

A130534 → D²∘x (shift +1)

A163932 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 A163932
\(P_{n}(x) = \left(3\,x + 3\,n\right)\,P_{n-1}(x) + \left(-1 + 3\,x - 3\,x^{2} + 3\,n - 6\,n x - 3\,n^{2}\right)\,P_{n-2}(x) + \left(-1 + 3\,x - 3\,x^{2} + x^{3} + 3\,n - 6\,n x + 3\,n x^{2} - 3\,n^{2} + 3\,n^{2} x + n^{3}\right)\,P_{n-3}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 3\,x\)
\(P_{2}(x) = 11 + 18\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1336
21118635
3501056010225
4274675510150151624
517644872441017503152113132
613068393964061419600483058828118124
710958435437240370422449068040114661008361172700

Show all 15 stored rows

Generated b-files: A000399 (row sums), A163932 ?

Computed in your browser ?

Technical details
Node id
n-2c8bb380cc75972b · revision 1 · JSON
Definition source
oeis_catalog:A163932
Fingerprint
f6-86ccdfaf0e2feef5

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:        3,        3;
  2:       11,       18,        6;
  3:       50,      105,       60,       10;
  4:      274,      675,      510,      150,       15;
  5:     1764,     4872,     4410,     1750,      315,      21;
  6:    13068,    39396,    40614,    19600,     4830,     588,     28;
  7:   109584,   354372,   403704,   224490,    68040,   11466,   1008,    36;
  8:  1026576,  3518100,  4342080,  2693250,   949095,  198450,  24360,  1620,   45;
  9: 10628640, 38260728, 50457000, 34169300, 13530825, 3313233, 508200, 47520, 2475, 55;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+3*x)*P(n-1,x) - (3*n^2-3*n+1+(6*n-3)*x+3*x^2)*P(n-2,x) + (n^3-3*n^2+3*n-1+(3*n^2-6*n+3)*x+(3*n-3)*x^2+x^3)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 3*x, P(2,x) = 11 + 18*x + 6*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[2, x_] := 11 + 18*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[(3*n + 3*x)*P[n - 1, x] - (3*n^2 - 3*n + 1 + (6*n - 3)*x + 3*x^2)*P[n - 2, x] + (n^3 - 3*n^2 + 3*n - 1 + (3*n^2 - 6*n + 3)*x + (3*n - 3)*x^2 + x^3)*P[n - 3, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 3*x; v[3] = 11 + 18*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = (3*n + 3*x)*v[i - 1] - (3*n^2 - 3*n + 1 + (6*n - 3)*x + 3*x^2)*v[i - 2] + (n^3 - 3*n^2 + 3*n - 1 + (3*n^2 - 6*n + 3)*x + (3*n - 3)*x^2 + x^3)*v[i - 3]); 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.

A163932: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:        1;
   2:        3,        3;
   3:       11,       18,        6;
   4:       50,      105,       60,       10;
   5:      274,      675,      510,      150,       15;
   6:     1764,     4872,     4410,     1750,      315,      21;
   7:    13068,    39396,    40614,    19600,     4830,     588,     28;
   8:   109584,   354372,   403704,   224490,    68040,   11466,   1008,    36;
   9:  1026576,  3518100,  4342080,  2693250,   949095,  198450,  24360,  1620,   45;
  10: 10628640, 38260728, 50457000, 34169300, 13530825, 3313233, 508200, 47520, 2475, 55;
%F Formula
The row polynomials satisfy P(n,x) = (3*n-3+3*x)*P(n-1,x) - (3*n^2-9*n+7+(6*n-9)*x+3*x^2)*P(n-2,x) + (n^3-6*n^2+12*n-8+(3*n^2-12*n+12)*x+(3*n-6)*x^2+x^3)*P(n-3,x) for n >= 4, with P(1,x) = 1, P(2,x) = 3 + 3*x, P(3,x) = 11 + 18*x + 6*x^2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + 3*x;
P[3, x_] := 11 + 18*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[(3*n - 3 + 3*x)*P[n - 1, x] - (3*n^2 - 9*n + 7 + (6*n - 9)*x + 3*x^2)*P[n - 2, x] + (n^3 - 6*n^2 + 12*n - 8 + (3*n^2 - 12*n + 12)*x + (3*n - 6)*x^2 + x^3)*P[n - 3, 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; v[2] = 3 + 3*x; v[3] = 11 + 18*x + 6*x^2; for(n = 4, N, my(i = n); v[i] = (3*n - 3 + 3*x)*v[i - 1] - (3*n^2 - 9*n + 7 + (6*n - 9)*x + 3*x^2)*v[i - 2] + (n^3 - 6*n^2 + 12*n - 8 + (3*n^2 - 12*n + 12)*x + (3*n - 6)*x^2 + x^3)*v[i - 3]); 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-337faa044bd16e1a
  • 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-337faa044bd16e1a
  • 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-337faa044bd16e1a
  • 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-337faa044bd16e1a

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

No candidate equalities or related sequences.

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