A105861
a(n) = (n/2) * Sum_{k=0..n} binomial(n,k)/gcd(n,k).
Original entry on oeis.org
1, 3, 10, 23, 76, 102, 442, 695, 1792, 2828, 11254, 13334, 53236, 65418, 155110, 347319, 1114096, 1259328, 4980718, 6223148, 15033700, 27548678, 96468970, 108761942, 352992576, 529504212, 1381165192, 2314603370, 7784628196
Offset: 1
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f[n_] := n*Sum[ Binomial[n, k] / GCD[n, k], {k, 0, n}]/2; Table[ f[n], {n, 30}]
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a(n) = sum(k=0, n, binomial(n, k)/gcd(n, k))*n/2; \\ Michel Marcus, Oct 19 2019
A105863
a(n) = n * Sum_{d|n} (binomial(n,d) / gcd(n,d)).
Original entry on oeis.org
1, 6, 12, 44, 30, 252, 56, 856, 846, 3080, 132, 20616, 182, 49532, 52110, 237232, 306, 1227096, 380, 4106320, 2470272, 15525092, 552, 86092176, 1328900, 270424752, 126624006, 1157002616, 870, 5577100260, 992, 19572325728, 6386892930
Offset: 1
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f[n_] := Block[{d = Divisors[n]}, n*Plus @@ (Binomial[n, d])]; Table[ f[n], {n, 34}]
A141680
Triangle read by rows: T(n,m) = (n/m)*binomial(n,m) if m divides n, otherwise 0.
Original entry on oeis.org
1, 4, 1, 9, 0, 1, 16, 12, 0, 1, 25, 0, 0, 0, 1, 36, 45, 40, 0, 0, 1, 49, 0, 0, 0, 0, 0, 1, 64, 112, 0, 140, 0, 0, 0, 1, 81, 0, 252, 0, 0, 0, 0, 0, 1, 100, 225, 0, 0, 504, 0, 0, 0, 0, 1
Offset: 1
1;
4, 1;
9, 0, 1;
16, 12, 0, 1;
25, 0, 0, 0, 1;
36, 45, 40, 0, 0, 1;
49, 0, 0, 0, 0, 0, 1;
64, 112, 0, 140, 0, 0, 0, 1;
81, 0, 252, 0, 0, 0, 0, 0, 1;
100, 225, 0, 0, 504, 0, 0, 0, 0, 1;
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t[n_, m_] = If[Mod[n, m] == 0, n/m, 0]*Binomial[n, m]; Table[Table[t[n, m], {m, 1, n}], {n, 1, 10}]; Flatten[%]
A134774
G.f.: A(x) = Product_{n>=1} G(x^n,n)^n where G(x,n) = 1 + x*G(x,n)^n.
Original entry on oeis.org
1, 1, 3, 6, 15, 26, 66, 110, 253, 460, 966, 1680, 3732, 6304, 13073, 23539, 47548, 82362, 171463, 293578, 597934, 1056830, 2105424, 3654919, 7533609, 12915780, 26112978, 46033557, 92504870, 160298673, 330468463, 568239653, 1161488784
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 26*x^5 + 66*x^6 +...
G.f.: A(x) = 1/(1-x) * G(x^2,2)^2 * G(x^3,3)^3 * G(x^4,4)^4 *...
where the functions G(x,n) are g.f.s of well-known sequences:
G(x,2) = g.f. of A000108 = 1 + x*G(x,2)^2;
G(x,3) = g.f. of A001764 = 1 + x*G(x,3)^3;
G(x,4) = g.f. of A002293 = 1 + x*G(x,4)^4 ; etc.
Explicitly, the product yielding the g.f. A(x) begins:
A(x) = [1 + x + x^2 + x^3 +...] * [1 + 2*x^2 + 5*x^4 + 14*x^6 +...] * [1 + 3*x^3 + 12*x^6 + 55*x^9 +...] * [1 + 4*x^4 + 22*x^8 + 140*x^12 +...] * ...
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a(n)=if(n==0,1, polcoeff(exp(sum(m=1,n, x^m*sumdiv(m,d, binomial(m,d)/gcd(m,d)))),n))
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a(n)=polcoeff(prod(m=1,n,(1/x*serreverse(x/(1+x^m +x*O(x^n))))^m),n)
Showing 1-4 of 4 results.
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