A206152
a(n) = Sum_{k=0..n} binomial(n,k)^(n+k).
Original entry on oeis.org
1, 2, 10, 326, 64066, 111968752, 1091576358244, 106664423412770932, 67305628532703785062402, 329378455047908259704557301276, 15577435010841058543979449475481629020, 4149966977623235242137197627437116176363522092
Offset: 0
L.g.f.: L(x) = 2*x + 10*x^2/2 + 326*x^3/3 + 64066*x^4/4 + 111968752*x^5/5 +...
where exponentiation yields A206151:
exp(L(x)) = 1 + 2*x + 7*x^2 + 120*x^3 + 16257*x^4 + 22426576*x^5 +...
Illustration of initial terms:
a(1) = 1^1 + 1^2 = 2;
a(2) = 1^2 + 2^3 + 1^4 = 10;
a(3) = 1^3 + 3^4 + 3^5 + 1^6 = 326;
a(4) = 1^4 + 4^5 + 6^6 + 4^7 + 1^8 = 64066; ...
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Table[Sum[Binomial[n,k]^(n+k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)
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{a(n)=sum(k=0,n,binomial(n,k)^(n+k))}
A206154
a(n) = Sum_{k=0..n} binomial(n,k)^(k+2).
Original entry on oeis.org
1, 2, 10, 110, 2386, 125752, 14921404, 3697835668, 2223231412546, 3088517564289836, 9040739066816429380, 63462297965044771663708, 1064766030857977088480630740, 37863276208844960432962611293828, 3144384748384240804260912067907833280
Offset: 0
L.g.f.: L(x) = 2*x + 10*x^2/2 + 110*x^3/3 + 2386*x^4/4 + 125752*x^5/5 +...
where exponentiation yields A206151:
exp(L(x)) = 1 + 2*x + 7*x^2 + 48*x^3 + 693*x^4 + 26632*x^5 + 2542514*x^6 +...
Illustration of initial terms:
a(1) = 1^2 + 1^3 = 2;
a(2) = 1^2 + 2^3 + 1^4 = 10;
a(3) = 1^2 + 3^3 + 3^4 + 1^5 = 110;
a(4) = 1^2 + 4^3 + 6^4 + 4^5 + 1^6 = 2386;
a(5) = 1^2 + 5^3 + 10^4 + 10^5 + 5^6 + 1^7 = 125752; ...
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Table[Sum[Binomial[n,k]^(k+2),{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jan 16 2014 *)
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{a(n)=sum(k=0,n,binomial(n,k)^(k+2))}
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 +...
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{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),", "))
A206155
G.f.: exp( Sum_{n>=1} A206156(n)*x^n/n ), where A206156(n) = Sum_{k=0..n} binomial(n,k)^(2*k).
Original entry on oeis.org
1, 2, 5, 38, 1425, 283002, 448468978, 2707673843860, 67018498701021670, 14506787732148113566364, 13603174532364904984495776225, 43960529641219941452921634596223366, 1207327102995668834632770987833295579308107, 188859837731175560954429490131760211759694331013582
Offset: 0
G.f.: A(x) = 1 + 2*x + 5*x^2 + 38*x^3 + 1425*x^4 + 283002*x^5 +...
where the logarithm of the g.f. begins:
log(A(x)) = 2*x + 6*x^2/2 + 92*x^3/3 + 5410*x^4/4 + 1400652*x^5/5 + 2687407464*x^6/6 +...+ A206156(n)*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^(2*k-0))+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 +...
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{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 1-5 of 5 results.
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