A336713
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.
Original entry on oeis.org
1, 1, 1, 6, 76, 1447, 37206, 1212194, 47975271, 2238595055, 120453255172, 7347494056729, 501273291296174, 37833413358907566, 3130557361463956074, 281854137496597897755, 27433898122963009937892, 2870816347095046227070383, 321430790732030793454519088
Offset: 0
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a[0] = 1; a[n_] := Sum[(-1)^(n - k) * Binomial[n, k] * Binomial[n + (n - 1)*k, k - 1], {k, 1, n}] / n; Array[a, 19, 0] (* Amiram Eldar, Aug 01 2020 *)
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{a(n) = if(n==0, 1, sum(k=1, n, (-1)^(n-k)*binomial(n, k)*binomial(n+(n-1)*k, k-1))/n)}
A336714
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-2)^(n-k) * binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.
Original entry on oeis.org
1, 1, 0, 2, 36, 766, 20910, 707472, 28740656, 1367040950, 74645106114, 4606416653654, 317237242964840, 24130334401571972, 2009783477119978508, 181958565624827141256, 17796032244661580019904, 1870078875109869688744870, 210155525478346375059816234, 25151873422906866362758095642
Offset: 0
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a[0] = 1; a[n_] := Sum[(-2)^(n - k) * Binomial[n, k] * Binomial[n + (n - 1)*k, k - 1], {k, 1, n}] / n; Array[a, 20, 0] (* Amiram Eldar, Aug 01 2020 *)
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{a(n) = if(n==0, 1, sum(k=1, n, (-2)^(n-k)*binomial(n, k)*binomial(n+(n-1)*k, k-1))/n)}
A336706
Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 4, 1, 1, 4, 11, 14, 9, 1, 1, 5, 20, 45, 42, 21, 1, 1, 6, 32, 113, 197, 132, 51, 1, 1, 7, 47, 234, 688, 903, 429, 127, 1, 1, 8, 65, 424, 1854, 4404, 4279, 1430, 323, 1, 1, 9, 86, 699, 4159, 15490, 29219, 20793, 4862, 835
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, ...
2, 5, 11, 20, 32, 47, 65, ...
4, 14, 45, 113, 234, 424, 699, ...
9, 42, 197, 688, 1854, 4159, 8192, ...
21, 132, 903, 4404, 15490, 43097, 101538, ...
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T[0, k_] := 1; T[n_, k_] := Sum[Binomial[n, j] * Binomial[n + (k - 1)*j, j - 1], {j, 1, n}] / n; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 01 2020 *)
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{T(n, k) = if(n==0, 1, sum(j=1, n, binomial(n, j)*binomial(n+(k-1)*j, j-1))/n)}
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{T(n, k) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k/(1-x*A)); polcoef(A, n)}
A336712
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} 2^(n-k) * binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.
Original entry on oeis.org
1, 1, 4, 30, 364, 6502, 158034, 4921112, 187897728, 8519286854, 447829041358, 26796275824186, 1798936842255128, 133933302810144684, 10953460639289615412, 976226180855018504472, 94181146038753255120480, 9778885058353578446996934, 1087326670244362420301889926
Offset: 0
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a[0] = 1; a[n_] := Sum[2^(n - k) * Binomial[n, k] * Binomial[n + (n - 1)*k, k - 1], {k, 1, n}] / n; Array[a, 19, 0] (* Amiram Eldar, Aug 01 2020 *)
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{a(n) = if(n==0, 1, sum(k=1, n, 2^(n-k)*binomial(n, k)*binomial(n+(n-1)*k, k-1))/n)}
Showing 1-4 of 4 results.