A206830
G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) * x^k ).
Original entry on oeis.org
1, 1, 2, 5, 34, 520, 14397, 993806, 222547738, 98753510701, 66772601607218, 82150206439975648, 310163020349941301606, 3022167582612808506550780, 47176617497043375266215814522, 1129578055293824008530028604347686, 62478430488069985838347598494293429802
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 34*x^4 + 520*x^5 + 14397*x^6 + ...
such that, by definition, the logarithm equals:
log(A(x)) = x*(1+x) + x^2*(1 + 6*x + x^2)/2 + x^3*(1 + 84*x + 84*x^2 + x^3)/3 + x^4*(1 + 1820*x + 12870*x^2 + 1820*x^3 + x^4)/4 + x^5*(1 + 53130*x + 3268760*x^2 + 3268760*x^3 + 53130*x^4 + x^5)/5 + ... + x^n/n*Sum_{k=0..n} A209330(n,k)*x^k + ...
More explicitly,
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 115*x^4/4 + 2416*x^5/5 + 83064*x^6/6 + ...
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{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m^2, m*k)*x^k)*x^m/m)+x*O(x^n)), n)}
for(n=0,15,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(""))
Showing 1-2 of 2 results.
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