A099237
a(n) = Sum_{k=0..n} binomial(n*(n-k), k).
Original entry on oeis.org
1, 1, 3, 10, 45, 251, 1624, 11908, 97545, 880660, 8664546, 92096731, 1050304775, 12778138842, 165033693175, 2253204163256, 32401745953105, 489207829112931, 7733130368443057, 127664099576228184, 2196149923000824756
Offset: 0
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A099237:= func< n | (&+[Binomial(n*j, n-j): j in [0..n]]) >;
[A099237(n): n in [0..30]]; // G. C. Greubel, Mar 09 2021
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A099237:= n-> add( binomial(n*j, n-j), j=0..n );
seq(A099237(n), n=0..30); # G. C. Greubel, Mar 09 2021
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Table[Sum[Binomial[n*(n - k), k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Feb 19 2018 *)
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def A099237(n): return sum( binomial(n*j, n-j) for j in (0..n))
[A099237(n) for n in (0..30)] # G. C. Greubel, Mar 09 2021
A157117
Triangle T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A008292(n*k + 1, n-k) if k <= n otherwise A008292(n*(n-k), k), read by rows.
Original entry on oeis.org
2, 1, 1, 1, 8, 1, 1, 131, 131, 1, 1, 8204, 29216, 8204, 1, 1, 2097187, 44136233, 44136233, 2097187, 1, 1, 2147483736, 846839476071, 503464582368, 846839476071, 2147483736, 1, 1, 8796093022411, 150092195453359483, 288159195861579519, 288159195861579519, 150092195453359483, 8796093022411, 1
Offset: 0
2;
1, 1;
1, 8, 1;
1, 131, 131, 1;
1, 8204, 29216, 8204, 1;
1, 2097187, 44136233, 44136233, 2097187, 1;
1, 2147483736, 846839476071, 503464582368, 846839476071, 2147483736, 1;
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Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >;
f:= func< n,k | k le n select Eulerian(n*k+1,n-k) else Eulerian(n*(n-k)+1, k) >;
A157117:= func< n,k | f(n,k) + f(n,n-k) >;
[A157117(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 11 2022
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f[n_, k_]:= If[k<=n, Eulerian[n*k+1,n-k], Eulerian[n*(n-k)+1,k]];
T[n_, k_]:= f[n,k] + f[n,n-k];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 11 2022 *)
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def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1))
def f(n,k): return Eulerian(n*k+1,n-k) if (kA157117(n,k): return f(n,k) + f(n,n-k)
flatten([[A157117(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 11 2022
A157118
Triangle T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A001263(n*k+1, n-k+1) if k <= n otherwise A001263(n*(n-k)+1, k+1) and T(1, k) = 1, read by rows.
Original entry on oeis.org
2, 1, 1, 1, 6, 1, 1, 27, 27, 1, 1, 88, 672, 88, 1, 1, 225, 9150, 9150, 225, 1, 1, 486, 98385, 395352, 98385, 486, 1, 1, 931, 1126951, 11748681, 11748681, 1126951, 931, 1, 1, 1632, 14600320, 402703120, 588593280, 402703120, 14600320, 1632, 1, 1, 2673, 201755880, 16093941435, 32251030119, 32251030119, 16093941435, 201755880, 2673, 1
Offset: 0
Triangle begins as:
2;
1, 1;
1, 6, 1;
1, 27, 27, 1;
1, 88, 672, 88, 1;
1, 225, 9150, 9150, 225, 1;
1, 486, 98385, 395352, 98385, 486, 1;
1, 931, 1126951, 11748681, 11748681, 1126951, 931, 1;
1, 1632, 14600320, 402703120, 588593280, 402703120, 14600320, 1632, 1;
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A001263:= func< n,k | Binomial(n-1,k-1)*Binomial(n,k)/(n-k+1) >;
f:= func< n,k | k le n select A001263(n*k+1,n-k+1) else A001263(n*(n-k)+1, k+1) >;
A157118:= func< n,k | n eq 1 select 1 else f(n,k) + f(n,n-k) >;
[A157118(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 11 2022
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A001263[n_, k_]:= Binomial[n-1,k-1]*Binomial[n,k]/(n-k+1);
f[n_, k_]:= If[k<=n, A001263[n*k+1,n-k+1], A001263[n*(n-k)+1,k+1]];
T[n_, k_]:= If[n==1, 1, f[n,k] + f[n,n-k]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 11 2022 *)
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def A001263(n,k): return binomial(n-1,k-1)*binomial(n,k)/(n-k+1)
def f(n,k): return A001263(n*k+1,n-k+1) if (kA001263(n*(n-k)+1, k+1)
def A157118(n,k): return 1 if (n==1) else f(n,k) + f(n,n-k)
flatten([[A157118(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 11 2022
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