basis-map coefficients of A113214, shifted → basis A013610

OEIS matches ?

No OEIS matches.

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:   -2,    -3;
  2:    4,    12,      9;
  3:   -8,   -36,    -54,     -27;
  4:   16,    96,    216,     216,      81;
  5:  -32,  -240,   -720,   -1080,    -810,    -243;
  6:   64,   576,   2160,    4320,    4860,    2916,     729;
  7: -128, -1344,  -6048,  -15120,  -22680,  -20412,  -10206,   -2187;
  8:  256,  3072,  16128,   48384,   90720,  108864,   81648,   34992,    6561;
  9: -512, -6912, -41472, -145152, -326592, -489888, -489888, -314928, -118098, -19683;

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

P[0, x_] := 1;
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] = 1; 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-e876d74ed42231b4
• 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-e876d74ed42231b4
• 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-e876d74ed42231b4

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-e876d74ed42231b4

  Geometric basis map (elementary) polytool ?

  • premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image reversal, shift 0 e-60aa79917842b999

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • basis_map A007318 e-c62e3171974f4f9c

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • parent_provers periodic_product e-5a4f04ff49b735c1 root: theorem check

              Parent-row provers of polytool polytool ?

            • preserver_image mobius, shift 0 e-4adca88d3fcf673e

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ parent_provers periodic_product e-5a4f04ff49b735c1 (expanded above)
        • parent_provers periodic_product e-c76ca1e6cd597d3b root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image affine, shift 0 e-c58069640929a62b

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers periodic_product e-5a4f04ff49b735c1 (expanded above)
            • ↵ preserver_image mobius, shift 0 e-4adca88d3fcf673e (expanded above)
    • ↵ basis_map A007318 e-c62e3171974f4f9c (expanded above)
    • ↵ parent_provers periodic_product e-c76ca1e6cd597d3b (expanded above)
    • ↵ preserver_image affine, shift 0 e-c58069640929a62b (expanded above)

Related sequences ?

Same fingerprint (23)

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

• A344565 → basis A013610
• A196347 → basis A013610
• A038338 → basis A013610
• A038208 → basis A013610
• A038221 → basis A013610
• A156992 → basis A013610
• A028326 → basis A013610
• A038325 → basis A013610
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
• A085880 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
• A120910 → x ↦ x/(x+1)
• A178821 → basis A013610
• A038220 → D
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
• A038243 → basis A013611
• A356546 → basis A013610
• A085881 → basis A013610
• A178822 → basis A013610
• A120910 → basis A013609
• A135838 → basis A013610
• A178820 → basis A013610
• A013620 → D