A038330 → basis A013609

OEIS matches ?

• A009968: 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,            256,              64;
  1:           4096,           6144,            3072,             512;
  2:          65536,         131072,           98304,           32768,            4096;
  3:        1048576,        2621440,         2621440,         1310720,          327680,           32768;
  4:       16777216,       50331648,        62914560,        41943040,        15728640,         3145728,          262144;
  5:      268435456,      939524096,      1409286144,      1174405120,       587202560,       176160768,        29360128,        2097152;
  6:     4294967296,    17179869184,     30064771072,     30064771072,     18790481920,      7516192768,      1879048192,      268435456,       16777216;
  7:    68719476736,   309237645312,    618475290624,    721554505728,    541165879296,    270582939648,     90194313216,    19327352832,     2415919104,     134217728;
  8:  1099511627776,  5497558138880,  12369505812480,  16492674416640,  14431090114560,   8658654068736,   3607772528640,  1030792151040,   193273528320,   21474836480,   1073741824;
  9: 17592186044416, 96757023244288, 241892558110720, 362838837166080, 362838837166080, 253987186016256, 126993593008128, 45354854645760, 11338713661440, 1889785610240, 188978561024, 8589934592;

The row polynomials satisfy P(n,x) = (16+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 256*x + 64*x^2.

P[0, x_] := 256 + 256*x + 64*x^2;
P[n_, x_] := P[n, x] = Expand[(16 + 8*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 + 256*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 8*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-6efc8fa5932928ce
• 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-6efc8fa5932928ce
• 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-6efc8fa5932928ce

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-6efc8fa5932928ce

  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 (67)

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

• A178820 → x ↦ x/(x+1)
• A130123 → basis A013609
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
• A178821 → basis A007318
• basis-map coefficients of A112227 → basis A013609
• A085881 → basis A013611
• A178821 → basis A013611
• A159854 → x ↦ x/(x+1)
• A178820 → basis A013611
• A120909 → basis A013610
• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-…
• A007318 → ×(1+2n+n²) → x ↦ x+1
• A028326 → basis A007318
• A038243 → basis A013610
• A178821 → x ↦ x/(x+1)
• A356546 → basis A013611
• A120909 → D
• A344565 → basis A013611
• A085880 → x ↦ x/(x+1)
• A135838 → x ↦ x+1
• A178822 → basis A013611
• A344565 → x ↦ x/(x+1)
• A038338 → basis A007318
• A135838 → x ↦ x/(x+1)
• A038208 → basis A013611
• A196347 → basis A007318
• basis-map coefficients of A021012 → basis A013609
• A038207 → D
• A085880 → x ↦ x+1
• A085881 → basis A007318
• A007318 → D → x ↦ x/(x+1)
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maxi…
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
• A013609 → basis A013610
• A135838 → basis A013611
• A028326 → x ↦ x/(x+1)
• basis-map coefficients of A113214, shifted → basis A013611
• A038208 → x ↦ x/(x+1)
• A156992 → basis A013611
• A196347 → basis A013611
• A178822 → x ↦ x/(x+1)
• A038221 → basis A013611
• A085880 → basis A013611
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x/(x+1)
• basis-map coefficients of A030528, shifted → basis A013609
• basis-map coefficients of A026729, shifted → basis A013609
• A178822 → x ↦ x+1
• A027466 → basis A013611
• A120910 → D
• A344565 → basis A007318
• basis-map coefficients of A026729 → basis A013609
• A028326 → basis A013611
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
• A038338 → basis A013611
• A038325 → basis A007318
• A156992 → basis A007318
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• A178820 → basis A007318
• A013609 → D
• A356546 → basis A007318
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x+1
• basis-map coefficients of A113214, shifted → x ↦ x+1
• A038325 → basis A013611
• A007318 → ×(1+2n+n²) → x ↦ x/(x+1)
• A038208 → basis A007318
• A136572 → 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