basis-map coefficients of A113214, shifted → basis A013611

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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,    -4;
  2:    4,    16,     16;
  3:   -8,   -48,    -96,     -64;
  4:   16,   128,    384,     512,      256;
  5:  -32,  -320,  -1280,   -2560,    -2560,    -1024;
  6:   64,   768,   3840,   10240,    15360,    12288,     4096;
  7: -128, -1792, -10752,  -35840,   -71680,   -86016,   -57344,   -16384;
  8:  256,  4096,  28672,  114688,   286720,   458752,   458752,   262144,    65536;
  9: -512, -9216, -73728, -344064, -1032192, -2064384, -2752512, -2359296, -1179648, -262144;

The row polynomials satisfy P(n,x) = -(2+4*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 + 4*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 + 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 - 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-6ef42593efb6357c
• 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-6ef42593efb6357c
• 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-6ef42593efb6357c

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-6ef42593efb6357c

  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 (67)

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

• A120909 → D
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
• A178821 → basis A013611
• A028326 → basis A013611
• A344565 → basis A013611
• A178820 → basis A007318
• A135838 → x ↦ x/(x+1)
• A085880 → x ↦ x/(x+1)
• A178820 → basis A013611
• A135838 → x ↦ x+1
• A038243 → basis A013610
• A356546 → basis A007318
• A120910 → D
• A135838 → basis A013611
• A038208 → x ↦ x/(x+1)
• basis-map coefficients of A021012 → basis A013609
• A038221 → basis A013611
• A013609 → basis A013610
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
• A178821 → basis A007318
• A038207 → D
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x/(x+1)
• A085880 → x ↦ x+1
• A007318 → ×(1+2n+n²) → x ↦ x/(x+1)
• A028326 → basis A007318
• basis-map coefficients of A026729 → basis A013609
• A178821 → x ↦ x/(x+1)
• A196347 → basis A013611
• A178822 → basis A013611
• A007318 → ×(1+2n+n²) → x ↦ x+1
• basis-map coefficients of A030528, shifted → basis A013609
• A027466 → basis A013611
• A038208 → basis A013611
• A159854 → x ↦ x/(x+1)
• A085880 → basis A013611
• A085881 → basis A013611
• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-…
• A178820 → x ↦ x/(x+1)
• A085881 → basis A007318
• A038330 → basis A013609
• A038208 → basis A007318
• A136572 → basis A013609
• A038338 → basis A007318
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x+1
• A013609 → D
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maxi…
• A028326 → x ↦ x/(x+1)
• A178822 → x ↦ x/(x+1)
• A156992 → basis A013611
• A356546 → basis A013611
• A156992 → basis A007318
• A038325 → basis A013611
• A130123 → basis A013609
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
• basis-map coefficients of A113214, shifted → x ↦ x+1
• basis-map coefficients of A026729, shifted → basis A013609
• A038338 → basis A013611
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• A178822 → x ↦ x+1
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• A038325 → basis A007318
• A344565 → x ↦ x/(x+1)
• A196347 → basis A007318
• basis-map coefficients of A112227 → basis A013609
• A120909 → basis A013610
• A007318 → D → x ↦ x/(x+1)
• A344565 → basis A007318