A159854 → basis A013609

OEIS matches ?

• A002063: 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 ?
• A183354: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A055841: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,    4,     4;
  1:   2,   10,    16,      8;
  2:   4,   24,    52,     48,     16;
  3:   8,   56,   152,    200,    128,     32;
  4:  16,  128,   416,    704,    656,    320,     64;
  5:  32,  288,  1088,   2240,   2720,   1952,    768,    128;
  6:  64,  640,  2752,   6656,   9920,   9344,   5440,   1792,    256;
  7: 128, 1408,  6784,  18816,  33152,  38528,  29568,  14464,   4096,   512;
  8: 256, 3072, 16384,  51200, 103936, 143360, 136192,  88064,  37120,  9216,  1024;
  9: 512, 6656, 38912, 135168, 310272, 494592, 559104, 448512, 250368, 92672, 20480, 2048;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1 + 4*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 2*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-4ff089327fda7415
• 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-4ff089327fda7415
• 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-4ff089327fda7415

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-4ff089327fda7415

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Riordan array (1-x,x/(1-x)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-b62074ba45057e82 root: theorem check

      Parent-row provers of polytool polytool ?

Related sequences ?

Same fingerprint (2)

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

• A391533: The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
• A391533 → reverse