basis-map coefficients of A026729, shifted → basis A013610

OEIS matches ?

• A262710: row sums, 15 terms agree (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:
   1:  1;
   2: -1,  -3;
   3:  1,   6,    9;
   4: -1,  -9,  -27,   -27;
   5:  1,  12,   54,   108,     81;
   6: -1, -15,  -90,  -270,   -405,   -243;
   7:  1,  18,  135,   540,   1215,   1458,    729;
   8: -1, -21, -189,  -945,  -2835,  -5103,  -5103,  -2187;
   9:  1,  24,  252,  1512,   5670,  13608,  20412,  17496,   6561;
  10: -1, -27, -324, -2268, -10206, -30618, -61236, -78732, -59049, -19683;

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

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(1 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = -(1 + 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, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, 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-fe4cc1f9dd1d3096
• Coefficients of one sign. Every nonzero row, n >= 1: 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-fe4cc1f9dd1d3096
• Hadamard kernel (Malo). Rows P_n, n >= 1, 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-fe4cc1f9dd1d3096

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-fe4cc1f9dd1d3096

  Geometric basis map (elementary) polytool ?

  • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-070da67d980ebd97 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-91ba1ede5d434b70

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-070da67d980ebd97 (expanded above)

Related sequences ?

Same fingerprint (16)

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

• A027465 → D
• A038279 → basis A013610
• A038243 → basis A013609
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
• A120910 → basis A007318
• basis-map coefficients of A030528, shifted → basis A013610
• A136572 → basis A013610
• A027465: Cube of lower triangular normalized binomial matrix.
• basis-map coefficients of A112227 → basis A013610
• basis-map coefficients of A026729 → basis A013610
• A013610 → basis A013611
• A013610 → D
• basis-map coefficients of A021012 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
• A130123 → basis A013610
• A120909 → x ↦ x/(x+1)