A206848
G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2) ).
Original entry on oeis.org
1, 2, 5, 53, 3422, 826606, 1335470713, 9548109569885, 190076214495558260, 18558289189760778318731, 10286810587274357297985552184, 16301371794177939084545371104827679, 91249944361047494534207504939405352235731, 3283593155431496336538359592977826684908598341441
Offset: 0
G.f.: A(x) = 1 + 2*x + 5*x^2 + 53*x^3 + 3422*x^4 + 826606*x^5 + 1335470713*x^6 +...
where the logarithm of the g.f. yields the l.g.f. of A206849:
log(A(x)) = 2*x + 6*x^2/2 + 137*x^3/3 + 13278*x^4/4 + 4098627*x^5/5 +...
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{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m^2,k^2))*x^m/m)+x*O(x^n)), n)}
for(n=0, 25, print1(a(n), ", "))
A206847
a(n) = Sum_{k=0..n} binomial(n^2, k^2) * binomial(n^2, (n-k)^2).
Original entry on oeis.org
1, 2, 18, 2270, 3678482, 51789416252, 9723940840418814, 13783866167176942874214, 260749663122506218247699587346, 35385577627626083328957267246097557212, 64138056102285851525440919122006580387539950268, 814449089808478655249485968539593253265395820497817710866
Offset: 0
L.g.f.: L(x) = 2*x + 18*x^2/2 + 2270*x^3/3 + 3678482*x^4/4 + 51789416252*x^5/5 +...
where exponentiation yields the g.f. of A206846:
exp(L(x)) = 1 + 2*x + 11*x^2 + 776*x^3 + 921193*x^4 + 10359730908*x^5 +...
Illustration of terms: by definition,
a(1) = C(1,0)*C(1,1) + C(1,1)*C(1,0);
a(2) = C(4,0)*C(4,4) + C(4,1)*C(4,1) + C(4,4)*C(4,0);
a(3) = C(9,0)*C(9,9) + C(9,1)*C(9,4) + C(9,4)*C(9,1) + C(9,9)*C(9,0);
a(4) = C(16,0)*C(16,16) + C(16,1)*C(16,9) + C(16,4)*C(16,4) + C(16,9)*C(16,1) + C(16,16)*C(16,0); ...
Numerically, the above evaluates to be:
a(1) = 1*1 + 1*1 = 2;
a(2) = 1*1 + 4*4 + 1*1 = 18;
a(3) = 1*1 + 9*126 + 126*9 + 1*1 = 2270;
a(4) = 1*1 + 16*11440 + 1820*1820 + 11440*16 + 1*1 = 3678482;
a(5) = 1*1 + 25*2042975 + 12650*2042975 + 2042975*12650 + 2042975*25 + 1*1 = 51789416252; ...
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Table[Sum[Binomial[n^2, k^2] * Binomial[n^2, (n-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^2,(n-k)^2)*binomial(n^2,k^2))}
for(n=1, 20, print1(a(n), ", "))
Showing 1-2 of 2 results.
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