A085880 → (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:
   1:    1;
   2:    1,      4;
   3:    2,     16,      18;
   4:    5,     60,     135,      80;
   5:   14,    224,     756,     896,      350;
   6:   42,    840,    3780,    6720,     5250,     1512;
   7:  132,   3168,   17820,   42240,    49500,    28512,     6468;
   8:  429,  12012,   81081,  240240,   375375,   324324,   147147,    27456;
   9: 1430,  45760,  360360, 1281280,  2502500,  2882880,  1961960,   732160,  115830;
  10: 4862, 175032, 1575288, 6534528, 15315300, 22054032, 20011992, 11202048, 3544398, 486200;

The row polynomials satisfy (n+(3*n^2-4*n)*x+(n^3-2*n^2+n)*x^2)*P(n,x) = (4*n-6+(12*n^2-18*n)*x+(4*n^3+6*n^2-22*n+6)*x^2+(4*n^3-6*n^2)*x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 6 + (12*n^2 - 18*n)*x + (4*n^3 + 6*n^2 - 22*n + 6)*x^2 + (4*n^3 - 6*n^2)*x^3)*P[n - 1, x])/(n + (3*n^2 - 4*n)*x + (n^3 - 2*n^2 + n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n - 6 + (12*n^2 - 18*n)*x + (4*n^3 + 6*n^2 - 22*n + 6)*x^2 + (4*n^3 - 6*n^2)*x^3)*v[i - 1])/(n + (3*n^2 - 4*n)*x + (n^3 - 2*n^2 + n)*x^2)); 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-0eee3767fe33ed2e
• 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-0eee3767fe33ed2e
• 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-0eee3767fe33ed2e
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-0eee3767fe33ed2e

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-0eee3767fe33ed2e

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

  • premise (sequence): Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-68a5cf53bdad0f8f root: theorem check

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

    • preserver_image reversal, shift 0 e-93440eb5a87a1e1b

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-68a5cf53bdad0f8f (expanded above)

Related sequences ?

Same fingerprint (12)

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

• basis-map coefficients of A113214, shifted → (θ+1)²
• A038208 → (θ+1)²
• A007318 → (θ+1)² → reverse
• A135838 → (θ+1)²
• A028326 → (θ+1)²
• A178822 → (θ+1)²
• A344565 → (θ+1)²
• A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
• A178821 → (θ+1)²
• A007318 → ×(1+2n+n²) → (θ+1)²
• A007318 → D → (θ+1)²
• A178820 → (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.