basis-map coefficients of A030528, shifted → basis A013609

OEIS matches ?

• A352779: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A141413: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS

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:  1;
  1: -1,  -2;
  2:  1,   4,    4;
  3: -1,  -6,  -12,   -8;
  4:  1,   8,   24,   32,    16;
  5: -1, -10,  -40,  -80,   -80,   -32;
  6:  1,  12,   60,  160,   240,   192,    64;
  7: -1, -14,  -84, -280,  -560,  -672,  -448,  -128;
  8:  1,  16,  112,  448,  1120,  1792,  1792,  1024,   256;
  9: -1, -18, -144, -672, -2016, -4032, -5376, -4608, -2304, -512;

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

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(1 + 2*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] = -(1 + 2*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-13dcb978bba71940
• 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-13dcb978bba71940
• 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-13dcb978bba71940

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 A013609 e-13dcb978bba71940

  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)

Related sequences ?

Candidate equalities (1)

Definitions that agree on the stored rows without an all-n certificate.

• basis-map coefficients of A113214, shifted → x ↦ x/(x+1) (15 rows compared; no all-index equality certificate was found)

Same fingerprint (66)

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

• 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-…
• A085880 → basis A013611
• A038208 → basis A007318
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• A159854 → x ↦ x/(x+1)
• A038325 → basis A013611
• A120909 → D
• A038207 → D
• A344565 → x ↦ x/(x+1)
• A085880 → x ↦ x+1
• A196347 → basis A013611
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x/(x+1)
• A007318 → D → x ↦ x/(x+1)
• A178822 → x ↦ x/(x+1)
• A038221 → basis A013611
• basis-map coefficients of A021012 → basis A013609
• A356546 → basis A013611
• A120909 → basis A013610
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
• A038338 → basis A007318
• basis-map coefficients of A113214, shifted → basis A013611
• 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…
• A013609 → D
• basis-map coefficients of A113214, shifted → x ↦ x+1
• A085881 → basis A007318
• basis-map coefficients of A026729, shifted → basis A013609
• A156992 → basis A007318
• basis-map coefficients of A026729 → basis A013609
• A178820 → x ↦ x/(x+1)
• A013609 → basis A013610
• A028326 → x ↦ x/(x+1)
• A027466 → basis A013611
• A344565 → basis A013611
• A135838 → basis A013611
• A038208 → basis A013611
• A038330 → basis A013609
• A130123 → basis A013609
• A196347 → basis A007318
• A028326 → basis A013611
• A135838 → x ↦ x/(x+1)
• A007318 → ×(1+2n+n²) → x ↦ x/(x+1)
• A178820 → basis A013611
• A156992 → basis A013611
• A120910 → D
• A085880 → x ↦ x/(x+1)
• A178820 → basis A007318
• A178821 → basis A007318
• A038208 → x ↦ x/(x+1)
• A178821 → basis A013611
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• A356546 → basis A007318
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x+1
• A178822 → basis A013611
• A038338 → basis A013611
• A178821 → x ↦ x/(x+1)
• A135838 → x ↦ x+1
• A085881 → basis A013611
• A178822 → x ↦ x+1
• A136572 → basis A013609
• A028326 → basis A007318
• A344565 → basis A007318
• A038243 → basis A013610
• basis-map coefficients of A112227 → basis A013609
• A007318 → ×(1+2n+n²) → x ↦ x+1
• A038325 → basis A007318