A089627: T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.

OEIS matches ?

• A002426: 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 ?
• A391403: triangle, 56 terms agree, 14 full rows row lengths from the atlas (tabf) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 992 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;
   3: 1,  2;
   4: 1,  6;
   5: 1, 12,   6;
   6: 1, 20,  30;
   7: 1, 30,  90,   20;
   8: 1, 42, 210,  140;
   9: 1, 56, 420,  560,  70;
  10: 1, 72, 756, 1680, 630;

The row polynomials satisfy -(n-1)*P(n,x) = -(2*n-3)*P(n-1,x) + (n-2-(4*n-8)*x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 3)*P[n - 1, x] + (n - 2 - (4*n - 8)*x)*P[n - 2, x])/(-n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(2*n - 3)*v[i - 1] + (n - 2 - (4*n - 8)*x)*v[i - 2])/(-n + 1)); 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;
  2: 1,  2;
  3: 1,  6;
  4: 1, 12,   6;
  5: 1, 20,  30;
  6: 1, 30,  90,   20;
  7: 1, 42, 210,  140;
  8: 1, 56, 420,  560,  70;
  9: 1, 72, 756, 1680, 630;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - 1)*v[i - 1] + (n - 1 - (4*n - 4)*x)*v[i - 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 >= 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-4d3032ecd7423900
• 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-4d3032ecd7423900
• 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-4d3032ecd7423900

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 hadamard_multiplier, shift -1 e-4d3032ecd7423900

  Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

  • premise (sequence): Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polyn…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers paper_theorem e-94e95c2ea0369f05 root: theorem check

      Parent-row provers of polytool polytool ?

Used in 5 proof steps

• e-23d8897c84d2236f preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, trian…
• e-272eb8130991447b preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle…
• e-502ba5da9494e9e3 preserver_image (affine, shift 0) → shift_x_plus_one of T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle re…
• e-65103a54d767b0ce preserver_image (euler_polynomial, shift 0) → theta_plus_one of T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read…
• e-65da073036a5e795 preserver_image (derivative_polynomial, shift 0) → derivative of T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by …

Related sequences ?

Same fingerprint (2)

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

• A391403: Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= fl…
• A069466 → γ

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