A143491 → basis A008277

OEIS matches ?

• A052852: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:         12,           8,           1;
  1:         60,          60,          15,          1;
  2:        360,         480,         180,         24,          1;
  3:       2520,        4200,        2100,        420,         35,         1;
  4:      20160,       40320,       25200,       6720,        840,        48,        1;
  5:     181440,      423360,      317520,     105840,      17640,      1512,       63,       1;
  6:    1814400,     4838400,     4233600,    1693440,     352800,     40320,     2520,      80,      1;
  7:   19958400,    59875200,    59875200,   27941760,    6985440,    997920,    83160,    3960,     99,    1;
  8:  239500800,   798336000,   898128000,  479001600,  139708800,  23950080,  2494800,  158400,   5940,  120,   1;
  9: 3113510400, 11416204800, 14270256000, 8562153600, 2854051200, 570810240, 71351280, 5662800, 283140, 8580, 143, 1;

The row polynomials satisfy P(n,x) = (n+4+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 12 + 8*x + x^2.

P[0, x_] := 12 + 8*x + x^2;
P[n_, x_] := P[n, x] = Expand[(n + 4 + 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] = 12 + 8*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 4 + 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)

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-13b78624414ee2b1
• 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-13b78624414ee2b1
• 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-13b78624414ee2b1

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-13b78624414ee2b1

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

  • premise (sequence): Unsigned 2-Stirling numbers of the first kind.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-84f3431e657be6ed root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers root_domain e-25f97869157cba20 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (1)

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

• A136124 → basis A008277

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
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