A139548 → basis A013609

OEIS matches ?

• A009976: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:         1;
  1:         8,         16,           8;
  2:        64,        256,         384,          256,           64;
  3:       512,       3072,        7680,        10240,         7680,          3072,           512;
  4:      4096,      32768,      114688,       229376,       286720,        229376,        114688,         32768,          4096;
  5:     32768,     327680,     1474560,      3932160,      6881280,       8257536,       6881280,       3932160,       1474560,        327680,         32768;
  6:    262144,    3145728,    17301504,     57671680,    129761280,     207618048,     242221056,     207618048,     129761280,      57671680,      17301504,       3145728,        262144;
  7:   2097152,   29360128,   190840832,    763363328,   2099249152,    4198498304,    6297747456,    7197425664,    6297747456,    4198498304,    2099249152,     763363328,     190840832,      29360128,      2097152;
  8:  16777216,  268435456,  2013265920,   9395240960,  30534533120,   73282879488,  134351945728,  191931351040,  215922769920,  191931351040,  134351945728,   73282879488,   30534533120,    9395240960,   2013265920,    268435456,    16777216;
  9: 134217728, 2415919104, 20535312384, 109521666048, 410706247680, 1149977493504, 2491617902592, 4271344975872, 5873099341824, 6525665935360, 5873099341824, 4271344975872, 2491617902592, 1149977493504, 410706247680, 109521666048, 20535312384, 2415919104, 134217728;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(8 + 16*x + 8*x^2)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (8 + 16*x + 8*x^2)*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-65b163cdbb61d2ab
• 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-65b163cdbb61d2ab
• 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-65b163cdbb61d2ab

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-65b163cdbb61d2ab

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-4755dd2fd7289161 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (3)

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

• A034870: Even-numbered rows of Pascal's triangle.
• A034870 → basis A013609
• A139548: Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.