A038330 → basis A007318

OEIS matches ?

• A009964: 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:            256,            128,             16;
  1:           4096,           3072,            768,             64;
  2:          65536,          65536,          24576,           4096,            256;
  3:        1048576,        1310720,         655360,         163840,          20480,          1024;
  4:       16777216,       25165824,       15728640,        5242880,         983040,         98304,          4096;
  5:      268435456,      469762048,      352321536,      146800640,       36700160,       5505024,        458752,        16384;
  6:     4294967296,     8589934592,     7516192768,     3758096384,     1174405120,     234881024,      29360128,      2097152,       65536;
  7:    68719476736,   154618822656,   154618822656,    90194313216,    33822867456,    8455716864,    1409286144,    150994944,     9437184,     262144;
  8:  1099511627776,  2748779069440,  3092376453120,  2061584302080,   901943132160,  270582939648,   56371445760,   8053063680,   754974720,   41943040,   1048576;
  9: 17592186044416, 48378511622144, 60473139527680, 45354854645760, 22677427322880, 7937099563008, 1984274890752, 354334801920, 44291850240, 3690987520, 184549376, 4194304;

The row polynomials satisfy P(n,x) = (16+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 128*x + 16*x^2.

P[0, x_] := 256 + 128*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(16 + 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] = 256 + 128*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 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-7b68ca495e98f7ea
• 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-7b68ca495e98f7ea
• 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-7b68ca495e98f7ea

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-7b68ca495e98f7ea

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-016885d52a801213 root: theorem check

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

    • parent_provers periodic_product e-0a35d6a32dc0856a root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (14)

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

• A013611 → D
• A136572 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013611
• A027466 → basis A013609
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
• A038231 → D
• basis-map coefficients of A112227 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013611
• A027467 → basis A013611
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
• basis-map coefficients of A021012 → basis A013611
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• basis-map coefficients of A026729 → basis A013611
• A130123 → 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