A143491 → basis A013609

OEIS matches ?

• A001720: 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,          14,           4;
  1:         60,          94,          48,           8;
  2:        360,         684,         476,         144,         16;
  3:       2520,        5508,        4700,        1960,        400,         32;
  4:      20160,       49104,       48616,       25080,       7120,       1056,        64;
  5:     181440,      482256,      535752,      322952,     114240,      23744,      2688,       128;
  6:    1814400,     5185440,     6322032,     4301024,    1788304,     465920,     74368,      6656,      256;
  7:   19958400,    60668640,    79913232,    59955328,   28273392,    8701728,   1749888,    221952,    16128,     512;
  8:  239500800,   767940480,  1080296064,   879290400,  459191360,  160967520,  38402112,   6163200,   637440,   38400,  1024;
  9: 3113510400, 10462227840, 15579729792, 13591367328, 7728068480, 3010960480, 821162496, 156925824, 20613120, 1774080, 90112, 2048;

The row polynomials satisfy P(n,x) = (n+4+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 14*x + 4*x^2.

P[0, x_] := 12 + 14*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(n + 4 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 12 + 14*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 4 + 2*x)*v[i - 1]); 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-7cc986a9eefcf6a5
• 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-7cc986a9eefcf6a5
• 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-7cc986a9eefcf6a5

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 A013609 e-7cc986a9eefcf6a5

  Geometric basis map (elementary) polytool ?

  • 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 A013609

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