A244751 a(n) = Sum_{k=0..n} C(n,k) * (n + 2^k)^(n-k) * 2^(k^2).

1, 4, 41, 1062, 98609, 41449418, 76876688017, 598204174499998, 19069330205237985089, 2462229757725772974948882, 1280330698557681260125588062425, 2672924626047136512609733349657605334, 22366552104466938320449948223074809365901745

Offset

0,2

Links

Table of n, a(n) for n=0..12.

Formula

a(n) = Sum_{k=0..n} C(n,k) * n^(n-k) * (1 + 2^k)^k.

Example

E.g.f.: A(x) = 1 + 4*x + 41*x^2/2! + 1062*x^3/3! + 98609*x^4/4! + 41449418*x^5/5! +...
ILLUSTRATION OF INITIAL TERMS:
a(1) = (1+2^0)^1*2^0 + (1+2^1)^0*2^1 = 4;
a(2) = (2+2^0)^2*2^0 + 2*(2+2^1)^1*2^1 + (2+2^2)^0*2^4 = 41;
a(3) = (3+2^0)^3*2^0 + 3*(3+2^1)^2*2^1 + 3*(3+2^2)^1*2^4 + (3+2^3)^0*2^9 = 1062;
a(4) = (4+2^0)^4*2^0 + 4*(4+2^1)^3*2^1 + 6*(4+2^2)^2*2^4 + 4*(4+2^3)^1*2^9 + (4+2^4)^0*2^16 = 98609; ...
where we have the binomial identity:
a(1) = 1^1*(1+2^0)^0 + 1^1*(1+2^1)^1 = 4;
a(2) = 2^2*(1+2^0)^0 + 2*2^1*(1+2^1)^1 + 2^0*(1+2^2)^2 = 41;
a(3) = 3^3*(1+2^0)^0 + 3*3^2*(1+2^1)^1 + 3*3^1*(1+2^2)^2 + 3^0*(1+2^3)^3 = 1062;
a(4) = 4^4*(1+2^0)^0 + 4*4^3*(1+2^1)^1 + 6*4^2*(1+2^2)^2 + 4*4^1*(1+2^3)^3 + 4^0*(1+2^4)^4 = 98609; ...

Prog

(PARI) {a(n) = sum(k=0, n, binomial(n, k) * (n + 2^k)^(n-k) * 2^(k^2) )}
for(n=0, 15, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, binomial(n, k) * n^(n-k) * (1 + 2^k)^k )}
for(n=0, 15, print1(a(n), ", "))

Crossrefs

Cf. A244689.

Sequence in context: A193363 A284276 A327026 * A022515 A172496 A059730

Adjacent sequences: A244748 A244749 A244750 * A244752 A244753 A244754

Keyword

nonn

Author

Paul D. Hanna, Jul 05 2014

Status

approved