A028326 → basis A013610

OEIS matches ?

• A020699: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A095687: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A020729: 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:    2;
  1:    4,     6;
  2:    8,    24,    18;
  3:   16,    72,   108,     54;
  4:   32,   192,   432,    432,    162;
  5:   64,   480,  1440,   2160,   1620,    486;
  6:  128,  1152,  4320,   8640,   9720,   5832,   1458;
  7:  256,  2688, 12096,  30240,  45360,  40824,  20412,   4374;
  8:  512,  6144, 32256,  96768, 181440, 217728, 163296,  69984,  13122;
  9: 1024, 13824, 82944, 290304, 653184, 979776, 979776, 629856, 236196, 39366;

The row polynomials satisfy P(n,x) = (2+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 2.

P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(2 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 3*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-28b777b3c0b25332
• 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-28b777b3c0b25332
• 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-28b777b3c0b25332

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 A013610 e-28b777b3c0b25332

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-3a144d1716c39a39 root: theorem check

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

    • preserver_image reversal, shift 0 e-07b4f7e7f8bd760f

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-3a144d1716c39a39 (expanded above)
        • parent_provers periodic_product e-2114c364938d0ed1 root: theorem check

          Parent-row provers of polytool polytool ?

    • ↵ parent_provers periodic_product e-2114c364938d0ed1 (expanded above)

Related sequences ?

Same fingerprint (23)

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

• A196347 → basis A013610
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A156992 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A038325 → basis A013610
• basis-map coefficients of A113214, shifted → basis A013610
• A178821 → basis A013610
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A013620 → D
• A038221 → basis A013610
• A038338 → basis A013610
• A135838 → basis A013610
• A356546 → basis A013610
• A120910 → basis A013609
• A085880 → basis A013610
• A038220 → D
• A178822 → basis A013610
• A085881 → basis A013610
• A344565 → basis A013610
• A178820 → basis A013610
• A038208 → basis A013610
• A120910 → x ↦ x/(x+1)
• A038243 → 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