hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).

Matches ?

Row sums ? A081294, A004171, A081654

Submission text ?

Triangle begins:
   1: 1,   6,    1;
   2: 1,  15,   15,     1;
   3: 1,  28,   70,    28,      1;
   4: 1,  45,  210,   210,     45,      1;
   5: 1,  66,  495,   924,    495,     66,      1;
   6: 1,  91, 1001,  3003,   3003,   1001,     91,      1;
   7: 1, 120, 1820,  8008,  12870,   8008,   1820,    120,     1;
   8: 1, 153, 3060, 18564,  43758,  43758,  18564,   3060,   153,    1;
   9: 1, 190, 4845, 38760, 125970, 184756, 125970,  38760,  4845,  190,   1;
  10: 1, 231, 7315, 74613, 319770, 646646, 646646, 319770, 74613, 7315, 231, 1;

The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1-2*x+x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1 + 6*x + x^2, P(2,x) = 1 + 15*x + 15*x^2 + x^3.

P[1, x_] := 1 + 6*x + x^2;
P[2, x_] := 1 + 15*x + 15*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 - 2*x + x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1 + 6*x + x^2; v[2] = 1 + 15*x + 15*x^2 + x^3; for(n = 3, N, my(i = n); v[i] = (2 + 2*x)*v[i - 1] - (1 - 2*x + x^2)*v[i - 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)

Properties

• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, +∞)
  by Three-term recurrence criterion
• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0)
  by Three-term recurrence criterion, Liu–Wang theorem for three-term recurrences
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Three-term recurrence criterion, Affine, reversal and Möbius substitutions (elementary), Liu–Wang theorem for three-term recurrences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Three-term recurrence criterion, Affine, reversal and Möbius substitutions (elementary), Liu–Wang theorem for three-term recurrences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m ≥ 2
  by Three-term recurrence criterion
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
  by Liu–Wang theorem for three-term recurrences
• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Ore-algebra identity of two definitions polytool ? step › direct check
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).?
    • Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
• ↑ Three-term recurrence criterion (above)
• ↑ Liu–Wang theorem for three-term recurrences (above)
• Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Used in 13

• basis_map_A008277 of hadamard_factor parent of Triangle read by rows: t(n,m) = … Image under a basis map u^k -> P_k (A008277)
• basis_map_A013609 of hadamard_factor parent of Triangle read by rows: t(n,m) = … Image under a basis map u^k -> P_k (A013609)
• basis_map_A013610 of hadamard_factor parent of Triangle read by rows: t(n,m) = … Image under a basis map u^k -> P_k (A013610)
• shift_x_plus_one of hadamard_factor parent of Triangle read by rows: t(n,m) = b… Image under a basis map u^k -> P_k (A007318)
• basis_map_A013611 of hadamard_factor parent of Triangle read by rows: t(n,m) = … Image under a basis map u^k -> P_k (A013611)
• Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m). Hadamard product with a multiplier sequence (binom(n+1, k))
• theta_plus_one_squared of hadamard_factor parent of Triangle read by rows: t(n,… Image under a real-rootedness preserver (euler polynomial, shift 0)
• hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * b… Image under a real-rootedness preserver (reversal, shift 0)

Related ?

Same fingerprint ?

• A086645: Triangle read by rows: T(n, k) = binomial(2n, 2k).
• A034870 → Veronese 2

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023