basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).

OEIS matches ?

• A004830: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A004438: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A000027: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
• A001477: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A004441: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A002377: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A005496: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A002837: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:   0,    2;
  2:  -3,    3,     3;
  3:   0,  -12,    12,    4;
  4:  10,  -20,   -20,   30,     5;
  5:   0,   60,  -120,    0,    60,     6;
  6: -35,  105,   105, -385,   105,   105,    7;
  7:   0, -280,   840, -280,  -840,   392,  168,    8;
  8: 126, -504,  -504, 3276, -2394, -1260, 1008,  252,   9;
  9:   0, 1260, -5040, 3360,  7560, -8820, -840, 2160, 360, 10;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 + 3*n + 1)*x*v[i - 1] + (4*n^2 + 4*n - (4*n^2 + 4*n)*x + (n^2 + n)*x^2)*v[i - 2])/(-n^2 - 2*n)); 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, 1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-676bb32d6a1b9c64
• 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-676bb32d6a1b9c64
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-72ab2c9b35ff70ef

Proof chains

2 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• preserver_image affine, shift 0 e-676bb32d6a1b9c64

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image euler_polynomial, shift 1 e-041eea8fc0a7af05

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

      • premise (sequence): narayana
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check three_term_recurrence e-46a461774e271ec3 root: theorem check

          Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

        • preserver_image reversal, shift 0 e-29ef477df4172e22

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): narayana
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check three_term_recurrence e-46a461774e271ec3 (expanded above)
    • basis_map A007318 e-a32917d44ac940a7

      Geometric basis map (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
        no recorded property covers this requirement
    • preserver_image reversal, shift 0 e-fcd77baa2d5e1258

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ preserver_image euler_polynomial, shift 1 e-041eea8fc0a7af05 (expanded above)
        • ↵ basis_map A007318 e-a32917d44ac940a7 (expanded above)
        • theorem_check liu_wang e-f8a41d41829f0ed1 root: theorem check

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

    • ↵ theorem_check liu_wang e-f8a41d41829f0ed1 (expanded above)
• theorem_check three_term_recurrence e-72ab2c9b35ff70ef root: theorem check

  Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Used in 4 proof steps

• e-a32917d44ac940a7 basis_map (A007318) → Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
• e-138e1ca38baf9fe5 preserver_image (affine, shift 0) → Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
• e-908b120a7c606a06 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(…
• e-d29f61265fea8d1e preserver_image (derivative_polynomial, shift 0) → derivative of basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*…

Related sequences ?

No candidate equalities or related sequences.

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023