A038319 → basis A007318

OEIS matches ?

• A009965: 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,            160,             25;
  1:           4096,           3840,           1200,            125;
  2:          65536,          81920,          38400,           8000,            625;
  3:        1048576,        1638400,        1024000,         320000,          50000,           3125;
  4:       16777216,       31457280,       24576000,       10240000,        2400000,         300000,         15625;
  5:      268435456,      587202560,      550502400,      286720000,       89600000,       16800000,       1750000,         78125;
  6:     4294967296,    10737418240,    11744051200,     7340032000,     2867200000,      716800000,     112000000,      10000000,       390625;
  7:    68719476736,   193273528320,   241591910400,   176160768000,    82575360000,    25804800000,    5376000000,     720000000,     56250000,     1953125;
  8:  1099511627776,  3435973836800,  4831838208000,  4026531840000,  2202009600000,   825753600000,  215040000000,   38400000000,   4500000000,   312500000,    9765625;
  9: 17592186044416, 60473139527680, 94489280512000, 88583700480000, 55364812800000, 24222105600000, 7569408000000, 1689600000000, 264000000000, 27500000000, 1718750000, 48828125;

The row polynomials satisfy P(n,x) = (16+5*x)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 160*x + 25*x^2.

P[0, x_] := 256 + 160*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[(16 + 5*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 + 160*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 5*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-eb35900dd666694f
• 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-eb35900dd666694f
• 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-eb35900dd666694f

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-eb35900dd666694f

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-f6c78a0c1a54b2b2 root: theorem check

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

    • parent_provers periodic_product e-97cca34dfbf5cc39 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.

• A013611 → 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