A178820 → (θ+1)²

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:   4,   16;
  2:  10,   80,    90;
  3:  20,  240,   540,    320;
  4:  35,  560,  1890,   2240,    875;
  5:  56, 1120,  5040,   8960,   7000,   2016;
  6:  84, 2016, 11340,  26880,  31500,  18144,   4116;
  7: 120, 3360, 22680,  67200, 105000,  90720,  41160,   7680;
  8: 165, 5280, 41580, 147840, 288750, 332640, 226380,  84480,  13365;
  9: 220, 7920, 71280, 295680, 693000, 997920, 905520, 506880, 160380, 22000;

The row polynomials satisfy (n+(3*n^2-n)*x+n^3*x^2)*P(n,x) = (n+3+(3*n^2+12*n+9)*x+(n^3+8*n^2+18*n+9)*x^2+(n^3+5*n^2+7*n+3)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + 3 + (3*n^2 + 12*n + 9)*x + (n^3 + 8*n^2 + 18*n + 9)*x^2 + (n^3 + 5*n^2 + 7*n + 3)*x^3)*P[n - 1, x])/(n + (3*n^2 - n)*x + n^3*x^2)]];
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] = ((n + 3 + (3*n^2 + 12*n + 9)*x + (n^3 + 8*n^2 + 18*n + 9)*x^2 + (n^3 + 5*n^2 + 7*n + 3)*x^3)*v[i - 1])/(n + (3*n^2 - n)*x + n^3*x^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-1cd2e816e298ff36
• 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-1cd2e816e298ff36
• 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-1cd2e816e298ff36
• 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-1cd2e816e298ff36

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-1cd2e816e298ff36

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

  • premise (sequence): Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-387009a21707feb6 root: theorem check

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

    • preserver_image reversal, shift 0 e-17b3b03bdb024482

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-387009a21707feb6 (expanded above)

Related sequences ?

Same fingerprint (12)

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

• A344565 → (θ+1)²
• A007318 → ×(1+2n+n²) → (θ+1)²
• basis-map coefficients of A113214, shifted → (θ+1)²
• A178821 → (θ+1)²
• A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
• A007318 → (θ+1)² → reverse
• A178822 → (θ+1)²
• A135838 → (θ+1)²
• A038208 → (θ+1)²
• A028326 → (θ+1)²
• A085880 → (θ+1)²
• A007318 → D → (θ+1)²

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.