A226391
a(n) = Sum_{k=0..n} binomial(k*n, k).
Original entry on oeis.org
1, 2, 9, 103, 2073, 58481, 2101813, 91492906, 4671050401, 273437232283, 18046800575211, 1325445408799007, 107200425419863009, 9466283137384124247, 906151826270369213655, 93459630239922214535911, 10331984296666203358431361, 1218745075041575200343722415
Offset: 0
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[(&+[Binomial(n*j,j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Aug 31 2022
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Table[Sum[Binomial[k*n, k], {k, 0, n}], {n, 0, 20}]
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A226391(n):=sum(binomial(k*n,k), k,0,n); makelist(A226391(n),n,0,30); /* Martin Ettl, Jun 06 2013 */
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@CachedFunction
def A226391(n): return sum(binomial(n*j, j) for j in (0..n))
[A226391(n) for n in (0..30)] # G. C. Greubel, Aug 31 2022
A349470
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(k*n,n).
Original entry on oeis.org
1, 1, 5, 65, 1394, 40378, 1470972, 64575585, 3315911300, 194921240846, 12905391110105, 950172113032181, 77000666619646717, 6810514097879311450, 652810277600420281734, 67407087759052608218945, 7459157975936646185855880, 880616251774021869817185430
Offset: 0
a(1) = (1/1!) * (1) = 1.
a(2) = (1/2!) * (-1*2 + 3*4) = 5.
a(3) = (1/3!) * (1*2*3 - 4*5*6 + 7*8*9) = 65.
a(4) = (1/4!) * (-1*2*3*4 + 5*6*7*8 - 9*10*11*12 + 13*14*15*16) = 1394.
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a[n_] := Sum[(-1)^(n - k) * Binomial[k*n, n], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2021 *)
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a(n) = sum(k=0, n, (-1)^(n-k)*binomial(k*n, n));
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a(n) = sum(j=0, n, (-1)^(n-j)*prod(k=(j-1)*n+1, j*n, k))/n!;
A383121
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(n*k,k).
Original entry on oeis.org
1, 0, 3, 47, 1093, 33029, 1236781, 55325416, 2879987209, 171061709417, 11418368571721, 846230146390001, 68949300160035373, 6126085419697733567, 589470974371501065845, 61068847238080533844679, 6777270943578364524130321, 802138434294752321142680145
Offset: 0
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Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[n k, k], {k, 0, n}], {n, 0, 17}]
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a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(n*k,k)); \\ Michel Marcus, Apr 17 2025
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