A200536 → basis A007318

OEIS matches ?

• A001024: 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:        6,         7,         2;
  2:       36,        84,        73,         28,          4;
  3:      216,       756,      1098,        847,        366,         84,          8;
  4:     1296,      6048,     12312,      14280,      10321,       4760,       1368,        224,         16;
  5:     7776,     45360,    118800,     183960,     186510,     129367,      62170,      20440,       4400,        560,         32;
  6:    46656,    326592,   1045872,    2026080,    2644380,    2449692,    1651609,     816564,     293820,      75040,      12912,      1344,        64;
  7:   279936,   2286144,   8654688,   20130768,   32140584,   37260972,   32346258,   21360031,   10782086,    4140108,    1190392,    248528,     35616,     3136,      128;
  8:  1679616,  15676416,  68491008,  185939712,  351068256,  488811456,  519185520,  429105936,  278905249,  143035312,   57687280,  18104128,   4334176,   765184,    93952,    7168,    256;
  9: 10077696, 105815808, 524040192, 1626428160, 3544969536, 5762225952, 7238929824, 7186557168, 5715544086, 3668760487, 1905181362, 798506352, 268108512, 71138592, 14588352, 2231040, 239616, 16128, 512;

The row polynomials satisfy P(n,x) = (6+7*x+2*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[(6 + 7*x + 2*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] = (6 + 7*x + 2*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-6e97878f95ac8f6d
• 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-6e97878f95ac8f6d
• 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-6e97878f95ac8f6d

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-6e97878f95ac8f6d

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-982ad459e6d62a8e 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.

• A200536 → basis A013610