A079638 → basis A008277

OEIS matches ?

• A025168: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A079621: triangle, 42 terms agree, 8 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?

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.

Triangle begins:
  0:          1;
  1:          4,          1;
  2:         24,         12,          1;
  3:        192,        144,         24,         1;
  4:       1920,       1920,        480,        40,         1;
  5:      23040,      28800,       9600,      1200,        60,        1;
  6:     322560,     483840,     201600,     33600,      2520,       84,      1;
  7:    5160960,    9031680,    4515840,    940800,     94080,     4704,    112,     1;
  8:   92897280,  185794560,  108380160,  27095040,   3386880,   225792,   8064,   144,   1;
  9: 1857945600, 4180377600, 2786918400, 812851200, 121927680, 10160640, 483840, 12960, 180, 1;

The row polynomials satisfy P(n,x) = -(2*n^2-2*n)*P(n-2,x) + (3*n+1+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.

P[0, x_] := 1;
P[1, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[-(2*n^2 - 2*n)*P[n - 2, x] + (3*n + 1 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = -(2*n^2 - 2*n)*v[i - 2] + (3*n + 1 + 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)

Triangle begins:
   1:          1;
   2:          4,          1;
   3:         24,         12,          1;
   4:        192,        144,         24,         1;
   5:       1920,       1920,        480,        40,         1;
   6:      23040,      28800,       9600,      1200,        60,        1;
   7:     322560,     483840,     201600,     33600,      2520,       84,      1;
   8:    5160960,    9031680,    4515840,    940800,     94080,     4704,    112,     1;
   9:   92897280,  185794560,  108380160,  27095040,   3386880,   225792,   8064,   144,   1;
  10: 1857945600, 4180377600, 2786918400, 812851200, 121927680, 10160640, 483840, 12960, 180, 1;

The row polynomials satisfy P(n,x) = -(2*n^2-6*n+4)*P(n-2,x) + (3*n-2+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + x.

P[1, x_] := 1;
P[2, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[-(2*n^2 - 6*n + 4)*P[n - 2, x] + (3*n - 2 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + x; for(n = 3, N, my(i = n); v[i] = -(2*n^2 - 6*n + 4)*v[i - 2] + (3*n - 2 + x)*v[i - 1] + x*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)

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-944c70f84abc9f7e
• 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-944c70f84abc9f7e
• 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-944c70f84abc9f7e

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

• basis_map A008277 e-944c70f84abc9f7e

  Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

  • premise (sequence): Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-84c88fc8b81d60d6 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

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. [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
3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023