A337510 a(n) = Sum_{k=0..n} T(n,k) where T(n,k) = (T(n-1, k-1) + T(n-1,k))^2.

1, 2, 6, 52, 3854, 21090612, 629815387162156, 561871511512925116799625359336, 446575758106416254441837050759254156476271759098752411181598

Offset: 0

Author

Glen Gilchrist, Aug 30 2020

Keywords

• easy
• nonn

Comments

Based on Pascal's triangle A007318 by additionally squaring the sum of each term generated. For example, in Pascal, n=3 gives 1,2,1. Here n=3 gives, 1^2, (1+1)^2, 1^2 = 1+4+1.

Examples

			1 = 1
1 + 1 = 2
1 + (1 + 1)^2  + 1 = 1 + 4 + 1 = 6
1 + (1 + 4)^2  + (4 + 1)^2 + 1 = 1 + 25 + 25 + 1 = 52
1 + (1 + 25)^2 + (25 + 25)^2 + (25 + 1)^2 + 1 = 1 + 676 + 2500 + 676 + 1 = 3854.
		

Crossrefs

Cf. A004019, A327563, A007318.

Programs

Python

def r(i):
  t = [[0, 1, 0], [0, 1, 1, 0]]
  for n in range(2, i+1):
    t.append([0])
    for k in range(1, n+2):
      t[n].append((t[n-1][k-1] + t[n-1][k])**2)
    t[n].append(0)
  return(sum(t[i]))

Formula

a(n) = Sum_{k=0..n} T(n,k) where T(n,k) = (T(n-1,k-1) + T(n-1,k))^2; T(0,0)=1; T(n,-1):=0; T(n,k):=0, n < k.