A207137
G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k*(n-k))*x^k ).
Original entry on oeis.org
1, 1, 2, 4, 17, 171, 3171, 101741, 7181615, 1274607729, 428568152553, 223160743256395, 185627109707405932, 320952534083059792786, 1367454166673309618606950, 11078799748881429582280609036, 137939599816546528357634500253053, 2679390013936303204526656964298150849
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 17*x^4 + 171*x^5 + 3171*x^6 +...
where the logarithm of the g.f. equals the l.g.f. of A207138:
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 51*x^4/4 + 761*x^5/5 + 17913*x^6/6 +...
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{a(n)=polcoeff(exp(sum(m=1,n,x^m/m*sum(k=0,m,binomial(m^2,k*(m-k))*x^k))+x*O(x^n)),n)}
for(n=0,25,print1(a(n),", "))
A228837
a(n) = Sum_{k=0..[n/2]} binomial((n-k)^2, (n-2*k)*k).
Original entry on oeis.org
1, 1, 2, 5, 38, 597, 14472, 554653, 44421258, 8933194659, 3408672951784, 1984802013951149, 1803179670478111304, 3323206887194925488269, 15156709454119350064982141, 132889643918499982093215167857, 1784438297905511051093397284187186
Offset: 0
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Table[Sum[Binomial[(n-k)^2, (n-2*k)*k],{k,0,Floor[n/2]}],{n,0,15}] (* Vaclav Kotesovec, Sep 05 2013 *)
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{a(n)=sum(k=0,n\2,binomial((n-k)^2, (n-2*k)*k))}
for(n=0,30,print1(a(n),", "))
A207140
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2,k^2).
Original entry on oeis.org
1, 2, 10, 407, 56746, 30771252, 115106662819, 1446405270234360, 53819202633553797290, 12313337704248075967333334, 12373818231445938048765251252260, 33156027144321617106970597265032233270, 409476940913917468665022448013012674533441891
Offset: 0
L.g.f.: L(x) = 2*x + 10*x^2/2 + 407*x^3/3 + 56746*x^4/4 + 30771252*x^5/5 +...
where exponentiation equals the g.f. of A207139:
exp(L(x)) = 1 + 2*x + 7*x^2 + 147*x^3 + 14481*x^4 + 6183605*x^5 +...
By definition, the initial terms begin: a(0) = 1;
a(1) = C(1,0)*C(1,0), + C(1,1)*C(1,1);
a(2) = C(2,0)*C(4,0), + C(2,1)*C(4,1), + C(2,2)*C(4,4);
a(3) = C(3,0)*C(9,0), + C(3,1)*C(9,1), + C(3,2)*C(9,4), + C(3,3)*C(9,9);
a(4) = C(4,0)*C(16,0), + C(4,1)*C(16,1), + C(4,2)*C(16,4), + C(4,3)*C(16,9), + C(4,4)*C(16,16); ...
which is evaluated as:
a(1) = 1*1 + 1*1 = 2;
a(2) = 1*1 + 2*4 + 1*1 = 10;
a(3) = 1*1 + 3*9 + 3*126 + 1*1 = 407;
a(4) = 1*1 + 4*16 + 6*1820 + 4*11440 + 1*1 = 56746;
a(5) = 1*1 + 5*25 + 10*12650 + 10*2042975 + 5*2042975 + 1*1 = 30771252;
a(6) = 1*1 + 6*36 + 15*58905 + 20*94143280 + 15*7307872110 + 6*600805296 + 1*1 = 115106662819; ...
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Table[Sum[Binomial[n,k] * Binomial[n^2,k^2], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)
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{a(n)=sum(k=0,n,binomial(n,k)*binomial(n^2,k^2))}
for(n=0,16,print1(a(n),", "))
Showing 1-3 of 3 results.
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