A139377 A Jacobsthal-Catalan triangle.

1, 1, 1, 3, 2, 1, 5, 6, 3, 1, 11, 15, 10, 4, 1, 21, 41, 30, 15, 5, 1, 43, 113, 92, 51, 21, 6, 1, 85, 327, 284, 171, 79, 28, 7, 1, 171, 982, 897, 570, 286, 115, 36, 8, 1, 341, 3066, 2895, 1913, 1016, 446, 160, 45, 9, 1

Offset

0,4

Comments

First column is A001045(n+1). Second column is A139379. Row sums are A139379(n+1).

Diagonal sums are A135582. Inverse of the Riordan array (1-x-x^2+4x^3-2x^4,x(1-x)).

Links

Table of n, a(n) for n=0..54.

P. Bala, A 4-parameter family of embedded Riordan arrays

Formula

Riordan array (1/(1-x-2x^2), xc(x)) where c(x) is the g.f. of A000108

From Peter Bala, Feb 20 2018: (Start)

Define a(n) = floor(2^(n+2)/3) - floor(2^(n+1)/3) = A001045(n+1). Then T(n,0) = a(n) and T(n,k) = Sum_{j = 0..n-k} a(j)*k/(2*n-k-2*j)*binomial(2*n-k-2*j,n-k-j) for 1 <= k <= n.

Define b(n) = (2/3)*(1+i)^(n-1) + (2/3)*(1-i)^(n-1) - (4/3)*(1+i)^(n-2) - (4/3)*(1-i)^(n-2) + (1/3)*(-1)^n*Fibonacci(n+1) + (2/3)*(-1)^n*Fibonacci(n). Then T(n,k) = Sum_{j = 0..n-k} b(j)*binomial(2*n-k-j,n) for 0 <= k <= n.

The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the rational function (1 - 2*x)/((1 - x)*(1 + x - x^2)*(1 - 2*x + 2*x^2)) * 1/(1 - x)^n about 0. For example, for n = 4, (1 - 2*x)/((1 - x)*(1 + x - x^2)*(1 - 2*x + 2*x^2)) * 1/(1 - x)^4 = (11*x^4 + 15*x^3 + 10*x^2 + 4*x + 1) + O(x^5). (End)

Example

Triangle begins
    1;
    1,   1;
    3,   2,   1;
    5,   6,   3,   1;
   11,  15,  10,   4,   1;
   21,  41,  30,  15,   5,   1;
   43, 113,  92,  51,  21,   6,   1;
   85, 327, 284, 171,  79,  28,   7,   1;
  171, 982, 897, 570, 286, 115,  36,   8,   1;
The production matrix for this array is
   1, 1,
   2, 1, 1,
  -2, 1, 1, 1,
   0, 1, 1, 1, 1,
   0, 1, 1, 1, 1, 1,
   0, 1, 1, 1, 1, 1, 1,
   0, 1, 1, 1, 1, 1, 1

Maple

#define auxiliary sequence
with(combinat):
b := proc (n)
  (2/3)*(1+I)^(n-1) + (2/3)*(1-I)^(n-1) - (4/3)*(1+I)^(n-2)-(4/3)*(1-I)^(n-2) + (1/3)*(-1)^n*fibonacci(n+1) + (2/3)*(-1)^n*fibonacci(n);
end proc:
A139377 := proc (n, k)
  add(b(j)*binomial(2*n-k-j, n), j = 0..n-k);
end proc:
#display sequence as a triangle
for n from 0 to 10 do
  seq(A139377(n, k), k = 0..n);
end do; # Peter Bala, Feb 20 2018

Crossrefs

Cf. A001045 (first column), A139379 (second column and row sums), A135582 (sums along shallow diagonals).

Sequence in context: A192022 A208608 A209577 * A368607 A138483 A110712

Adjacent sequences: A139374 A139375 A139376 * A139378 A139379 A139380

Keyword

easy,nonn,tabl

Author

Paul Barry, Apr 15 2008

Status

approved