A026374 → θ+1

OEIS matches ?

• A026950: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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,  6,   3;
  3: 1,  8,  12,   4;
  4: 1, 12,  33,  24,    5;
  5: 1, 14,  51,  68,   35,    6;
  6: 1, 18,  90, 180,  150,   54,    7;
  7: 1, 20, 117, 300,  375,  234,   70,   8;
  8: 1, 24, 174, 576,  975,  864,  406,  96,   9;
  9: 1, 26, 210, 808, 1695, 2034, 1414, 560, 117, 10;

The row polynomials satisfy P(n,x) = -(1+6*x+11*x^2+6*x^3+x^4)*P(n-4,x) + (2+6*x+2*x^2)*P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 6*x + 3*x^2, P(3,x) = 1 + 8*x + 12*x^2 + 4*x^3.

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 1 + 6*x + 3*x^2;
P[3, x_] := 1 + 8*x + 12*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-(1 + 6*x + 11*x^2 + 6*x^3 + x^4)*P[n - 4, x] + (2 + 6*x + 2*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 6*x + 3*x^2; v[4] = 1 + 8*x + 12*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(1 + 6*x + 11*x^2 + 6*x^3 + x^4)*v[i - 4] + (2 + 6*x + 2*x^2)*v[i - 2]); 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-9d3ba3f44f61794d
• 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-9d3ba3f44f61794d
• 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-9d3ba3f44f61794d

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

• preserver_image euler_polynomial, shift 0 e-9d3ba3f44f61794d

  Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-6e3ca670df3d0a74 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image reversal, shift 0 e-b2e3690a28625635

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = …
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-6e3ca670df3d0a74 (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.