A206156
a(n) = Sum_{k=0..n} binomial(n,k)^(2*k).
Original entry on oeis.org
1, 2, 6, 92, 5410, 1400652, 2687407464, 18947436116184, 536104663173431874, 130559883231879141946580, 136031455187223511721647272376, 483565526783420050082035900177878504, 14487924180895151383693101563813954330590756
Offset: 0
L.g.f.: L(x) = 2*x + 6*x^2/2 + 92*x^3/3 + 5410*x^4/4 + 1400652*x^5/5 +...
where exponentiation yields A206155:
exp(L(x)) = 1 + 2*x + 5*x^2 + 38*x^3 + 1425*x^4 + 283002*x^5 + 448468978*x^6 +...
Illustration of initial terms:
a(1) = 1^0 + 1^2 = 2;
a(2) = 1^0 + 2^2 + 1^4 = 6;
a(3) = 1^0 + 3^2 + 3^4 + 1^6 = 92;
a(4) = 1^0 + 4^2 + 6^4 + 4^6 + 1^8 = 5410;
a(5) = 1^0 + 5^2 + 10^4 + 10^6 + 5^8 + 1^10 = 1400652; ...
-
Table[Sum[Binomial[n,k]^(2*k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)
-
{a(n)=sum(k=0,n,binomial(n,k)^(2*k))}
for(n=0,16,print1(a(n),", "))
A206153
G.f.: exp( Sum_{n>=1} A206154(n)*x^n/n ), where A206154(n) = Sum_{k=0..n} binomial(n,k)^(k+2).
Original entry on oeis.org
1, 2, 7, 48, 693, 26632, 2542514, 533442978, 278979307990, 343728261289376, 904762216681139381, 5771110378770242683658, 88742047516327429085056353, 2912737209806573079629325613400, 224604736339682169442980060945290802
Offset: 0
G.f.: A(x) = 1 + 2*x + 7*x^2 + 48*x^3 + 693*x^4 + 26632*x^5 + 2542514*x^6 +...
where the logarithm of the g.f. begins:
log(A(x)) = 2*x + 10*x^2/2 + 110*x^3/3 + 2386*x^4/4 + 125752*x^5/5 + 14921404*x^6/6 +...+ A206154(n)*x^n/n +...
-
{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^(k+2))+x*O(x^n))),n)}
for(n=0,16,print1(a(n),", "))
A206157
G.f.: exp( Sum_{n>=1} A206158(n)*x^n/n ), where A206158(n) = Sum_{k=0..n} binomial(n,k)^(2*k+1).
Original entry on oeis.org
1, 2, 7, 102, 6261, 2423430, 6686021554, 61335432894584, 2941073857435300366, 1190520035262419577871332, 1696475310227140760623646031573, 9980324833243234634513255755001535870, 565171444566758371735408026461987217216896790
Offset: 0
G.f.: A(x) = 1 + 2*x + 7*x^2 + 102*x^3 + 6261*x^4 + 2423430*x^5 +...
where the logarithm of the g.f. begins:
log(A(x)) = 2*x + 10*x^2/2 + 272*x^3/3 + 24226*x^4/4 + 12053252*x^5/5 + 40086916024*x^6/6 +...+ A206158(n)*x^n/n +...
-
{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^(2*k+1))+x*O(x^n))),n)}
for(n=0,16,print1(a(n),", "))
Showing 1-3 of 3 results.
Comments