A038321 → basis A013611

OEIS matches ?

• A009990: 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,             1008,              784;
  1:           5832,            27216,            42336,             21952;
  2:         104976,           653184,          1524096,           1580544,             614656;
  3:        1889568,         14696640,         45722880,          71124480,           55319040,           17210368;
  4:       34012224,        317447424,       1234517760,        2560481280,         2987228160,         1858719744,          481890304;
  5:      612220032,       6666395904,      31109847552,       80655160320,       125463582720,       117099343872,        60718178304,        13492928512;
  6:    11019960576,     137137287168,     746636341248,     2322868617216,      4516688977920,      5620768505856,      4371708837888,      1942981705728,       377801998336;
  7:   198359290368,    2777030065152,   17279298183168,    62717452664832,    146340722884608,    227641124487168,    236072277245952,    157381518163968,     61203923730432,     10578455953408;
  8:  3570467226624,   55540601303040,  388784209121280,  1612734497095680,   4390221686538240,   8195080481538048,  10623252476067840,   9442891089838080,   5508353135738880,   1904122071613440,   296196766695424;
  9: 64268410079232, 1099703905800192, 8553252600668160, 39915178803118080, 124180556276367360, 270437655890755584, 420680798052286464, 467423108946984960, 363551306958766080, 188508085089730560, 58646959805693952, 8293509467471872;

The row polynomials satisfy P(n,x) = (18+28*x)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 1008*x + 784*x^2.

P[0, x_] := 324 + 1008*x + 784*x^2;
P[n_, x_] := P[n, x] = Expand[(18 + 28*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 + 1008*x + 784*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 28*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-e460f89b751823ed
• 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-e460f89b751823ed
• 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-e460f89b751823ed

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 A013611 e-e460f89b751823ed

  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 ?

No candidate equalities or related sequences.

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