A028326 → θ+1

OEIS matches ?

• A057711: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A129952: row sums, 15 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:
  0: 2;
  1: 2,  4;
  2: 2,  8,   6;
  3: 2, 12,  18,   8;
  4: 2, 16,  36,  32,   10;
  5: 2, 20,  60,  80,   50,   12;
  6: 2, 24,  90, 160,  150,   72,   14;
  7: 2, 28, 126, 280,  350,  252,   98,  16;
  8: 2, 32, 168, 448,  700,  672,  392, 128,  18;
  9: 2, 36, 216, 672, 1260, 1512, 1176, 576, 162, 20;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (n + 2)*x + (n + 1)*x^2)*v[i - 1])/(1 + n*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-421af44bc39d42c8
• 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-421af44bc39d42c8
• 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-421af44bc39d42c8
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-421af44bc39d42c8

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-421af44bc39d42c8

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

  • premise (sequence): Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-3a144d1716c39a39 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • preserver_image reversal, shift 0 e-07b4f7e7f8bd760f

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-3a144d1716c39a39 (expanded above)
        • parent_provers periodic_product e-2114c364938d0ed1 root: theorem check

          Parent-row provers of polytool polytool ?

    • ↵ parent_provers periodic_product e-2114c364938d0ed1 (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…
• A178820 → θ+1
• A028326 → D+1
• A178821 → θ+1
• A007318 → ×(1+2n+n²) → θ+1
• A135838 → D+1
• A344565 → θ+1
• A178820 → D+1
• A344565 → D+1
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• A007318 → ×(1+(3/2)n+(1/2)n²) → D+1
• A178822 → D+1
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• basis-map coefficients of A113214, shifted → D+1
• A178822 → θ+1
• A038208 → θ+1
• A178821 → D+1
• A135838 → θ+1
• A007318 → ×(1+2n+n²) → D+1
• A007318 → ×(1+(3/2)n+(1/2)n²) → θ+1
• A038208 → D+1
• A085880 → D+1
• A007318 → D → D+1
• basis-map coefficients of A113214, shifted → θ+1
• A085880 → θ+1
• A007318 → D → 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. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.