A038281 → basis A007318

OEIS matches ?

• A001023: 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:          121,           66,             9;
  1:         1331,         1089,           297,           27;
  2:        14641,        15972,          6534,         1188,           81;
  3:       161051,       219615,        119790,        32670,         4455,          243;
  4:      1771561,      2898918,       1976535,       718740,       147015,        16038,         729;
  5:     19487171,     37202781,      30438639,     13835745,      3773385,       617463,       56133,        2187;
  6:    214358881,    467692104,     446433372,    243509112,     83014470,     18112248,     2469852,      192456,       6561;
  7:   2357947691,   5787689787,    6313843404,   4017900348,   1643686506,    448278138,    81505116,     9526572,     649539,     19683;
  8:  25937424601,  70738430730,   86815346805,  63138434040,  30134252610,   9862119036,  2241390690,   349307640,   35724645,   2165130,   59049;
  9: 285311670611, 855935011833, 1167184107045, 954968814855, 520892080830, 198886067226, 54241654698, 10566556110, 1440894015, 130990365, 7144929, 177147;

The row polynomials satisfy P(n,x) = (11+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 66*x + 9*x^2.

P[0, x_] := 121 + 66*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(11 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 121 + 66*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 3*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-e643036831fc9f95
• 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-e643036831fc9f95
• 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-e643036831fc9f95

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-e643036831fc9f95

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-da334b7c58401d78 root: theorem check

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

    • parent_provers periodic_product e-c5656743efdffc45 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (2)

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

• A038303 → basis A013610
• A013618 → basis A013611

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