A228836
Triangle defined by T(n,k) = binomial(n^2, (n-k)*k), for n>=0, k=0..n, as read by rows.
Original entry on oeis.org
1, 1, 1, 1, 4, 1, 1, 36, 36, 1, 1, 560, 1820, 560, 1, 1, 12650, 177100, 177100, 12650, 1, 1, 376992, 30260340, 94143280, 30260340, 376992, 1, 1, 13983816, 8217822536, 92263734836, 92263734836, 8217822536, 13983816, 1, 1, 621216192, 3284214703056, 159518999862720, 488526937079580, 159518999862720, 3284214703056, 621216192, 1
Offset: 0
The triangle of coefficients C(n^2, (n-k)*k), n>=k, k=0..n, begins:
1;
1, 1;
1, 4, 1;
1, 36, 36, 1;
1, 560, 1820, 560, 1;
1, 12650, 177100, 177100, 12650, 1;
1, 376992, 30260340, 94143280, 30260340, 376992, 1;
1, 13983816, 8217822536, 92263734836, 92263734836, 8217822536, 13983816, 1;
...
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T[n_,k_]:=Binomial[n^2, (n-k)*k]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten (* Stefano Spezia, Aug 02 2025 *)
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{T(n,k)=binomial(n^2, (n-k)*k)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
A207138
L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k*(n-k))*x^k = Sum_{n>=1} a(n)*x^n/n.
Original entry on oeis.org
1, 3, 7, 51, 761, 17913, 688745, 56611987, 11405877739, 4272862207703, 2450039810788461, 2224842228379519641, 4169966883810355864393, 19139862395982576668262825, 166161479603614500915921996017, 2206856314384330228779059994929555
Offset: 1
L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 51*x^4/4 + 761*x^5/5 + 17913*x^6/6 +...
where exponentiation equals the g.f. of A207137:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 17*x^4 + 171*x^5 + 3171*x^6 +...
To illustrate the definition, the l.g.f. equals the series:
L(x) = (1 + x)*x + (1 + 4*x + 1*x^2)*x^2/2
+ (1 + 36*x + 36*x^2 + 1*x^3)*x^3/3
+ (1 + 560*x + 1820*x^2 + 560*x^3 + 1*x^4)*x^4/4
+ (1 + 12650*x + 177100*x^2 + 177100*x^3 + 12650*x^4 + 1*x^5)*x^5/5
+ (1 + 376992*x + 30260340*x^2 + 94143280*x^3 + 30260340*x^4 + 376992*x^5 + 1*x^6)*x^6/6 +...
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Table[n*Sum[Binomial[(n-k)^2, k*(n-2*k)]/(n-k),{k,0,Floor[n/2]}],{n,1,20}] (* Vaclav Kotesovec, Mar 04 2014 *)
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{a(n)=n*polcoeff(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m^2, k*(m-k))*x^k))+x*O(x^n),n)}
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{a(n)=n*sum(k=0,n\2,binomial((n-k)^2,k*(n-2*k))/(n-k))}
for(n=1,20,print1(a(n),", "))
Showing 1-2 of 2 results.
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