A192000 Sum of binomial numbers A000332(k+3), with k in the reduced residue system modulo n.

0, 1, 6, 16, 56, 71, 252, 296, 651, 721, 2002, 1282, 4368, 3402, 5782, 6672, 15504, 7947, 26334, 15702, 28868, 28457, 65780, 30212, 85580, 63063, 103284, 81452, 201376, 66102, 278256, 174624, 255794, 228684, 383166, 206838, 658008, 391419, 576394, 413244, 1086008

Offset: 1

Author

Wolfdieter Lang, Jun 22 2011

Keywords

• nonn
• easy

Comments

The reduced residue system modulo n used here is the set of numbers k from the set {0,1,...,n-1} which satisfy gcd(k,n)=1. There are phi(n) = A000010(n) such numbers k.

This is the m=4 member of a family of sequences, call them rmnS(m) (reduced mod n sum), with entries rmnS(m;n):=sum(binomial(k+m-1,m),0<=k<=n-1 with gcd(k,n)=1), m>=0, n>=1. Recall gcd(0,n)=n.

The members for m=0, 1, 2 and 3 are A000010, A023896, A127415, and A189918, respectively, where in the m=1 and 2 cases the offset for n=1 should be taken as 0 (not 1).

Examples

			a(6) = A000332(4) + A000292(8)= 1 + 70 = 71.
a(6) = (6/6!)*(6*3666*(1/3) + 5*137*2 - 182) = 71.
a(12) = A000332(4) + A000332(8) + A000332(10) + A000332(14) = 1 + 70 + 210 + 1001 = 1282.
a(12) = (12/6!)*(12*18258*(1/3) + 5*407*2 - 182) = 1282.
		

Crossrefs

Cf. A189918, A023896, A127415, A189922.

Programs

PARI

a(n) = sum(k=0, n-1, if (gcd(n,k) == 1, binomial(k+3, 4))); \\ Michel Marcus, Feb 01 2016

Formula

a(n) = sum(A000332(k+3), 0<=k<=n-1, gcd(k,n)=1), n>=1.

a(n) = (n/6!)*(n*(6*n^3+45*n^2+110*n+90)*P(1,n) + 5*(2*n^2+9*n+11)*P(-1,n) - P(-3,n)), n>=2, with P(k,n):= J(k,n)/n^k, where J(k,n) is the Jordan function (see A000010, A007434, A059376 - A059378, A069091 - A069095).

Extensions

More terms from Michel Marcus, Feb 01 2016