A085880 → D+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:
   1:     1;
   2:     2,      1;
   3:     6,      8,       2;
   4:    20,     45,      30,       5;
   5:    70,    224,     252,     112,      14;
   6:   252,   1050,    1680,    1260,     420,      42;
   7:   924,   4752,    9900,   10560,    5940,    1584,     132;
   8:  3432,  21021,   54054,   75075,   60060,   27027,    6006,    429;
   9: 12870,  91520,  280280,  480480,  500500,  320320,  120120,  22880,  1430;
  10: 48620, 393822, 1400256, 2858856, 3675672, 3063060, 1633632, 525096, 87516, 4862;

The row polynomials satisfy (n^2-n+n*x)*P(n,x) = (4*n^2-6*n+(4*n^2-2*n-6)*x+(4*n-6)*x^2)*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^2 - 6*n + (4*n^2 - 2*n - 6)*x + (4*n - 6)*x^2)*P[n - 1, x])/(n^2 - n + n*x)]];
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^2 - 6*n + (4*n^2 - 2*n - 6)*x + (4*n - 6)*x^2)*v[i - 1])/(n^2 - n + n*x)); 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-17a2449071a8dc49
• 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-17a2449071a8dc49
• 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-17a2449071a8dc49
• 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-17a2449071a8dc49

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 derivative_polynomial, shift 0 e-17a2449071a8dc49

  Laguerre’s theorem on differential operators [SP14] 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 (27)

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

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

References

1. [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
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.