A038321 → basis A007318

OEIS matches ?

• A009969: 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:            324,             252,              49;
  1:           5832,            6804,            2646,             343;
  2:         104976,          163296,           95256,           24696,            2401;
  3:        1889568,         3674160,         2857680,         1111320,          216090,           16807;
  4:       34012224,        79361856,        77157360,        40007520,        11668860,         1815156,          117649;
  5:      612220032,      1666598976,      1944365472,      1260236880,       490092120,       114354828,        14823774,         823543;
  6:    11019960576,     34284321792,     46664771328,     36294822144,     17643316320,      5489031744,      1067311728,      118590192,       5764801;
  7:   198359290368,    694257516288,   1079956136448,    979960197888,    571643448768,    222305785632,     57634833312,     9605805552,     933897762,     40353607;
  8:  3570467226624,  13885150325760,  24299013070080,  25198976517120,  17149303463040,   8003008282752,   2593567499040,   576348333120,   84050798580,   7263649260,   282475249;
  9: 64268410079232, 274925976450048, 534578287541760, 623674668798720, 485080297954560, 264099273330816, 102705272961984, 28529242489440, 5547352706280, 719101276740, 55930099302, 1977326743;

The row polynomials satisfy P(n,x) = (18+7*x)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 252*x + 49*x^2.

P[0, x_] := 324 + 252*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[(18 + 7*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 324 + 252*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 7*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-5a47242ff4b92549
• 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-5a47242ff4b92549
• 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-5a47242ff4b92549

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 A007318 e-5a47242ff4b92549

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-f6f8234ea600f6f8 root: theorem check

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

    • parent_provers periodic_product e-16634379058ff4dd 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.

• A013613 → basis A013610

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