basis-map coefficients of A112227 → basis A013609

OEIS matches ?

• A376478: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A155734: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A000244: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A133494: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A140429: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A027313: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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,  4,   4;
   2: 1,  6,  12,    8;
   3: 1,  8,  24,   32,   16;
   4: 1, 10,  40,   80,   80,    32;
   5: 1, 12,  60,  160,  240,   192,    64;
   6: 1, 14,  84,  280,  560,   672,   448,   128;
   7: 1, 16, 112,  448, 1120,  1792,  1792,  1024,   256;
   8: 1, 18, 144,  672, 2016,  4032,  5376,  4608,  2304,   512;
   9: 1, 20, 180,  960, 3360,  8064, 13440, 15360, 11520,  5120,  1024;
  10: 1, 22, 220, 1320, 5280, 14784, 29568, 42240, 42240, 28160, 11264, 2048;

The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 4*x + 4*x^2.

P[1, x_] := 1 + 4*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1 + 4*x + 4*x^2; for(n = 2, N, my(i = n); 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, 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-7bee75fc05394bb8
• 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-7bee75fc05394bb8
• 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-7bee75fc05394bb8

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-7bee75fc05394bb8

  Geometric basis map (elementary) polytool ?

  • premise (sequence): basis-map coefficients of A scaled Hermite triangle.
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-470643e70af29063 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-f9cce4eb6656235e

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of A scaled Hermite triangle.
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-470643e70af29063 (expanded above)

Related sequences ?

Same fingerprint (67)

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

• A178822 → x ↦ x+1
• A038330 → basis A013609
• basis-map coefficients of A113214, shifted → x ↦ x+1
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x/(x+1)
• A007318 → ×(1+2n+n²) → x ↦ x+1
• A120910 → D
• A356546 → basis A013611
• A178822 → x ↦ x/(x+1)
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• A196347 → basis A013611
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x+1
• A085880 → basis A013611
• A038208 → basis A007318
• A027466 → basis A013611
• A130123 → basis A013609
• A038325 → basis A013611
• A135838 → x ↦ x+1
• A178821 → basis A013611
• A085880 → x ↦ x/(x+1)
• A178822 → basis A013611
• A178821 → x ↦ x/(x+1)
• A028326 → x ↦ x/(x+1)
• A013609 → basis A013610
• A038325 → basis A007318
• A085881 → basis A013611
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
• basis-map coefficients of A026729, shifted → basis A013609
• A178820 → basis A013611
• A344565 → basis A013611
• A007318 → D → x ↦ x/(x+1)
• 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-…
• A038243 → basis A013610
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• A038338 → basis A013611
• A135838 → basis A013611
• A344565 → basis A007318
• A135838 → x ↦ x/(x+1)
• A178820 → basis A007318
• A120909 → D
• A085881 → basis A007318
• 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…
• A356546 → basis A007318
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
• basis-map coefficients of A026729 → basis A013609
• A178820 → x ↦ x/(x+1)
• A007318 → ×(1+2n+n²) → x ↦ x/(x+1)
• A038208 → x ↦ x/(x+1)
• A136572 → basis A013609
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
• A028326 → basis A007318
• A159854 → x ↦ x/(x+1)
• basis-map coefficients of A021012 → basis A013609
• A028326 → basis A013611
• A038338 → basis A007318
• A038221 → basis A013611
• A156992 → basis A007318
• A156992 → basis A013611
• A038207 → D
• A038208 → basis A013611
• A120909 → basis A013610
• A085880 → x ↦ x+1
• A178821 → basis A007318
• A196347 → basis A007318
• A013609 → D
• basis-map coefficients of A113214, shifted → basis A013611
• basis-map coefficients of A030528, shifted → basis A013609
• A344565 → x ↦ x/(x+1)