A167006
G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) ).
Original entry on oeis.org
1, 2, 6, 66, 4258, 1337374, 1933082159, 11353941470188, 291885138650054688, 29463501750534915665304, 12844314786465829040693498639, 21675661852919288704454219459892060, 156969579902607123047763327413679853875703
Offset: 0
G.f.: A(x) = 1 + 2*x + 6*x^2 + 66*x^3 + 4258*x^4 + 1337374*x^5 +...
log(A(x)) = 2*x + 8*x^2/2 + 170*x^3/3 + 16512*x^4/4 + 6643782*x^5/5 + 11582386286*x^6/6 +...+ A167009(n)*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m^2,k*m))*x^m/m)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))
A228904
Triangle defined by g.f. A(x,y) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n*k, k^2) * y^k ), as read by rows.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 4, 26, 62, 1, 1, 5, 70, 1087, 1031, 1, 1, 6, 155, 9257, 124702, 24782, 1, 1, 7, 301, 51397, 4479983, 26375325, 774180, 1, 1, 8, 532, 215129, 79666708, 5059028293, 8735721640, 29763855, 1, 1, 9, 876, 736410, 891868573, 357346615545, 10783389596184, 4162906254188, 1359654560, 1
Offset: 0
This triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 7, 1;
1, 4, 26, 62, 1;
1, 5, 70, 1087, 1031, 1;
1, 6, 155, 9257, 124702, 24782, 1;
1, 7, 301, 51397, 4479983, 26375325, 774180, 1;
1, 8, 532, 215129, 79666708, 5059028293, 8735721640, 29763855, 1;
1, 9, 876, 736410, 891868573, 357346615545, 10783389596184, 4162906254188, 1359654560, 1;
...
G.f.: A(x,y) = 1 + (1+y)*x + (1+2*y+y^2)*x^2 + (1+3*y+7*y^2+y^3)*x^3 + (1+4*y+26*y^2+62*y^3+y^4)*x^4 + (1+5*y+70*y^2+1087*y^3+1031*y^4+y^5)*x^5 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2*y + y^2)*x^2/2
+ (1 + 3*y + 15*y^2 + y^3)*x^3/3
+ (1 + 4*y + 70*y^2 + 220*y^3 + y^4)*x^4/4
+ (1 + 5*y + 210*y^2 + 5005*y^3 + 4845*y^4 + y^5)*x^5/5
+ (1 + 6*y + 495*y^2 + 48620*y^3 + 735471*y^4 + 142506*y^5 + y^6)*x^6/6 +...
in which the coefficients form A228832(n,k) = binomial(n*k, k^2).
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{T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, x^m/m*sum(j=0, m, binomial(m*j, j^2)*y^j))+x*O(x^n)), n, x), k, y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
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), ", "))
A228808
a(n) = Sum_{k=0..n} binomial(n*k, k^2).
Original entry on oeis.org
1, 2, 4, 20, 296, 10067, 927100, 219541877, 110728186648, 137502766579907, 448577320868198789, 3169529341990169816462, 51243646781214826181569316, 2201837465728010770618930322223, 215520476721579201896200887266792583, 45634827026091489574547858030506357191920
Offset: 0
L.g.f.: L(x) = 2*x + 4*x^2/2 + 20*x^3/3 + 296*x^4/4 + 10067*x^5/5 +...
where
exp(L(x)) = 1 + 2*x + 4*x^2 + 12*x^3 + 94*x^4 + 2195*x^5 + 158904*x^6 + 31681195*x^7 +...+ A228809(n)*x^n +...
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Table[Sum[Binomial[n*k, k^2],{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Sep 06 2013 *)
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a(n)=sum(k=0,n,binomial(n*k,k^2))
for(n=0,20,print1(a(n),", "))
Showing 1-4 of 4 results.
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