A038328 → basis A013609

OEIS matches ?

• A001027: 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:           196,            112,             16;
  1:          2744,           2352,            672,             64;
  2:         38416,          43904,          18816,           3584,           256;
  3:        537824,         768320,         439040,         125440,         17920,          1024;
  4:       7529536,       12907776,        9219840,        3512320,        752640,         86016,          4096;
  5:     105413504,      210827008,      180708864,       86051840,      24586240,       4214784,        401408,        16384;
  6:    1475789056,     3373232128,     3373232128,     1927561216,     688414720,     157351936,      22478848,      1835008,       65536;
  7:   20661046784,    53128406016,    60718178304,    40478785536,   17348050944,    4956585984,     944111616,    115605504,     8257536,     262144;
  8:  289254654976,   826441871360,  1062568120320,   809575710720,  404787855360,  138784407552,   33043906560,   5394923520,   578027520,   36700160,   1048576;
  9: 4049565169664, 12727204818944, 18181721169920, 15584332431360, 8905332817920, 3562133127168, 1017752322048, 207704555520, 29672079360, 2825912320, 161480704, 4194304;

The row polynomials satisfy P(n,x) = (14+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 112*x + 16*x^2.

P[0, x_] := 196 + 112*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(14 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 196 + 112*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 4*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-ccf7b0982517b62a
• 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-ccf7b0982517b62a
• 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-ccf7b0982517b62a

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-ccf7b0982517b62a

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-ca869591900be157 root: theorem check

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

    • parent_provers periodic_product e-48411a1ad7cd0ada 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.

• A038255 → 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