Real-rooted atlas

A079638: Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.

A079638 nonnegative coefficients

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

Recurrence from OEIS A079638
\(P_{n}(x) = \left(x + 3\,n\right)\,P_{n-1}(x) + \left(2\,n - 2\,n^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
2149124
39083181192
47448702753011920
5756010474427568545123040
69144014389270924148051435631322560
71285200223335612741963242894116026748415160960

Show all 15 stored rows

Generated b-files: A079638 ?

Computed in your browser ?

Technical details
Node id
n-6ace63f6e12c5046 · revision 1 · JSON
Definition source
oeis_catalog:A079638
Fingerprint
f6-bd19e0a95c26bd62

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,         1;
  2:        14,         9,         1;
  3:        90,        83,        18,         1;
  4:       744,       870,       275,        30,        1;
  5:      7560,     10474,      4275,       685,       45,       1;
  6:     91440,    143892,     70924,     14805,     1435,      63,      1;
  7:   1285200,   2233356,   1274196,    324289,    41160,    2674,     84,    1;
  8:  20603520,  38769840,  24870572,   7398972,  1151409,   98280,   4578,  108,   1;
  9: 371226240, 745785936, 526791060, 177945200, 32559975, 3419913, 209790, 7350, 135, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+x)*P(n-1,x) - (2*n^2-2*n)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(3*n + x)*P[n - 1, x] - (2*n^2 - 2*n)*P[n - 2, 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 + x; for(n = 2, N, my(i = n + 1); v[i] = (3*n + x)*v[i - 1] - (2*n^2 - 2*n)*v[i - 2]); 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.

A079638: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:         1;
   2:         3,         1;
   3:        14,         9,         1;
   4:        90,        83,        18,         1;
   5:       744,       870,       275,        30,        1;
   6:      7560,     10474,      4275,       685,       45,       1;
   7:     91440,    143892,     70924,     14805,     1435,      63,      1;
   8:   1285200,   2233356,   1274196,    324289,    41160,    2674,     84,    1;
   9:  20603520,  38769840,  24870572,   7398972,  1151409,   98280,   4578,  108,   1;
  10: 371226240, 745785936, 526791060, 177945200, 32559975, 3419913, 209790, 7350, 135, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3*n-3+x)*P(n-1,x) - (2*n^2-6*n+4)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(3*n - 3 + x)*P[n - 1, x] - (2*n^2 - 6*n + 4)*P[n - 2, 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 + x; for(n = 3, N, my(i = n); v[i] = (3*n - 3 + x)*v[i - 1] - (2*n^2 - 6*n + 4)*v[i - 2]); 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, +inf)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-fc633ad2be7d0c80
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-84c88fc8b81d60d6, e-fc633ad2be7d0c80
  • 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-84c88fc8b81d60d6
  • 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-84c88fc8b81d60d6
  • 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-84c88fc8b81d60d6

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 5 proof steps

No candidate equalities or related sequences.

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023