A135838 → θ+1

OEIS matches ?

No OEIS matches.

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:  2,   4;
  2:  2,   8,    6;
  3:  4,  24,   36,    16;
  4:  4,  32,   72,    64,    20;
  5:  8,  80,  240,   320,   200,    48;
  6:  8,  96,  360,   640,   600,   288,    56;
  7: 16, 224, 1008,  2240,  2800,  2016,   784,  128;
  8: 16, 256, 1344,  3584,  5600,  5376,  3136, 1024,  144;
  9: 32, 576, 3456, 10752, 20160, 24192, 18816, 9216, 2592, 320;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((2 + (2*n + 6)*x + (4*n + 6)*x^2 + (2*n + 2)*x^3)*v[i - 2])/(1 + (n - 1)*x)); 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-999c61cd00d73b61
• 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-999c61cd00d73b61
• 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-999c61cd00d73b61

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-999c61cd00d73b61

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

  • premise (sequence): Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image reversal, shift 0 e-07f7d31230240b98

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-51cb02553cf4a8ac root: theorem check

          Parent-row provers of polytool polytool ?

    • ↵ parent_provers periodic_product e-51cb02553cf4a8ac (expanded above)

Related sequences ?

Same fingerprint (27)

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

• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
• A178821 → D+1
• A085880 → D+1
• A038208 → θ+1
• A028326 → D+1
• A344565 → θ+1
• A007318 → D → D+1
• A007318 → ×(1+(3/2)n+(1/2)n²) → θ+1
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
• A007318 → D → θ+1
• A007318 → ×(1+2n+n²) → D+1
• basis-map coefficients of A113214, shifted → D+1
• A085880 → θ+1
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• A038208 → D+1
• basis-map coefficients of A113214, shifted → θ+1
• A178821 → θ+1
• A344565 → D+1
• A028326 → θ+1
• A178820 → D+1
• A135838 → D+1
• A007318 → ×(1+2n+n²) → θ+1
• A007318 → ×(1+(3/2)n+(1/2)n²) → D+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• A178822 → D+1
• A178822 → θ+1
• A178820 → θ+1

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.