basis-map coefficients of A026729, shifted → basis A013611

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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,  -4;
   3:  1,   8,   16;
   4: -1, -12,  -48,   -64;
   5:  1,  16,   96,   256,    256;
   6: -1, -20, -160,  -640,  -1280,   -1024;
   7:  1,  24,  240,  1280,   3840,    6144,    4096;
   8: -1, -28, -336, -2240,  -8960,  -21504,  -28672,  -16384;
   9:  1,  32,  448,  3584,  17920,   57344,  114688,  131072,   65536;
  10: -1, -36, -576, -5376, -32256, -129024, -344064, -589824, -589824, -262144;

The row polynomials satisfy P(n,x) = -(1+4*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 + 4*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 + 4*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-f6ed0d45ba75837f
• 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-f6ed0d45ba75837f
• 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-f6ed0d45ba75837f

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 A013611 e-f6ed0d45ba75837f

  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 (14)

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

• A038231 → D
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
• basis-map coefficients of A021012 → basis A013611
• A130123 → basis A013611
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• basis-map coefficients of A026729 → basis A013611
• A027467 → basis A013611
• A038330 → basis A007318
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
• basis-map coefficients of A112227 → basis A013611
• A027466 → basis A013609
• A136572 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013611
• A013611 → D