A280580: Triangle read by rows: T(n,k) = binomial(2*n,2*k)*binomial(2*n-2*k,n-k)/(n+1-k) for 0<=k<=n.

OEIS matches ?

• A026945: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A280580: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 0 ?

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:
   1:    1;
   2:    1,      1;
   3:    2,      6,       1;
   4:    5,     30,      15,       1;
   5:   14,    140,     140,      28,       1;
   6:   42,    630,    1050,     420,      45,      1;
   7:  132,   2772,    6930,    4620,     990,     66,     1;
   8:  429,  12012,   42042,   42042,   15015,   2002,    91,    1;
   9: 1430,  51480,  240240,  336336,  180180,  40040,  3640,  120,   1;
  10: 4862, 218790, 1312740, 2450448, 1837836, 612612, 92820, 6120, 153, 1;

The row polynomials satisfy -(2*n^2-n)*P(n,x) = -(8*n^2-16*n+6+(6*n^2-12*n+6)*x)*P(n-1,x) - ((16*n-12)*x-(4*n-3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^2 - 16*n + 6 + (6*n^2 - 12*n + 6)*x)*P[n - 1, x] - ((16*n - 12)*x - (4*n - 3)*x^2)*D[P[n - 1, x], x])/(-2*n^2 + n)]];
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] = (-(8*n^2 - 16*n + 6 + (6*n^2 - 12*n + 6)*x)*v[i - 1] - ((16*n - 12)*x - (4*n - 3)*x^2)*deriv(v[i - 1], x))/(-2*n^2 + n)); 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)

Triangle begins:
  0:    1;
  1:    1,      1;
  2:    2,      6,       1;
  3:    5,     30,      15,       1;
  4:   14,    140,     140,      28,       1;
  5:   42,    630,    1050,     420,      45,      1;
  6:  132,   2772,    6930,    4620,     990,     66,     1;
  7:  429,  12012,   42042,   42042,   15015,   2002,    91,    1;
  8: 1430,  51480,  240240,  336336,  180180,  40040,  3640,  120,   1;
  9: 4862, 218790, 1312740, 2450448, 1837836, 612612, 92820, 6120, 153, 1;

The row polynomials satisfy -(2*n^2+3*n+1)*P(n,x) = -(8*n^2-2+6*n^2*x)*P(n-1,x) - ((16*n+4)*x-(4*n+1)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^2 - 2 + 6*n^2*x)*P[n - 1, x] - ((16*n + 4)*x - (4*n + 1)*x^2)*D[P[n - 1, x], x])/(-2*n^2 - 3*n - 1)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(8*n^2 - 2 + 6*n^2*x)*v[i - 1] - ((16*n + 4)*x - (4*n + 1)*x^2)*deriv(v[i - 1], x))/(-2*n^2 - 3*n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 >= 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-7eafdf1af1515f93
• 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-7eafdf1af1515f93
• 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-7eafdf1af1515f93
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-7eafdf1af1515f93

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

• theorem_check liu_wang e-7eafdf1af1515f93 root: theorem check

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

Related sequences ?

No candidate equalities or related sequences.

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