A336709
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} (-2)^(n-j) * binomial(n,j) * binomial(n+(k-1)*j,j-1) for n > 0.
Original entry on oeis.org
1, 1, 1, 1, 1, -2, 1, 1, -1, 2, 1, 1, 0, -1, 4, 1, 1, 1, -1, 5, -24, 1, 1, 2, 2, 0, -3, 48, 1, 1, 3, 8, 5, 2, -21, 24, 1, 1, 4, 17, 36, 13, 0, 51, -464, 1, 1, 5, 29, 109, 177, 36, -5, 41, 1376, 1, 1, 6, 44, 240, 766, 922, 104, 0, -391, -704
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
-2, -1, 0, 1, 2, 3, 4, ...
2, -1, -1, 2, 8, 17, 29, ...
4, 5, 0, 5, 36, 109, 240, ...
-24, -3, 2, 13, 177, 766, 2177, ...
48, -21, 0, 36, 922, 5699, 20910, ...
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T[0, k_] := 1; T[n_, k_] := Sum[(-2)^(n - j) * 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, (-2)^(n-j)*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+2*x*A)); polcoef(A, n)}
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)}
A335871
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.
Original entry on oeis.org
1, 1, 3, 20, 234, 4159, 101538, 3182454, 122285201, 5575750271, 294529785168, 17697480642005, 1192398100081202, 89053864927236146, 7302988011333915878, 652439391227186881683, 63077327237347821501754, 6561701255914880362990927, 730833849063629052249986940
Offset: 0
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a[0] = 1; a[n_] := Sum[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, binomial(n, k)*binomial(n+(n-1)*k, k-1))/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)}
A336728
a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-n)^(n-k) * binomial(n,k) * binomial(n,k-1) for n > 0.
Original entry on oeis.org
1, 1, -1, 1, 9, -174, 2575, -38219, 588833, -9274418, 141253551, -1739881142, -753419447, 1379742127908, -83720072007585, 4059017293956301, -184613801568558975, 8254420480122200214, -369177108304219471457, 16608406418618863804990, -750673988803431836351799
Offset: 0
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a[0] = 1; a[n_] := Sum[(-n)^(n - k) * Binomial[n, k] * Binomial[n , k - 1], {k, 1, n}] / n; Array[a, 21, 0] (* Amiram Eldar, Aug 02 2020 *)
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{a(n) = if(n==0, 1, sum(k=1, n, (-n)^(n-k)*binomial(n, k)*binomial(n, k-1))/n)}
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{a(n) = sum(k=0, n, (-n)^k*(n+1)^(n-k)*binomial(n, k)*binomial(n+k, n)/(k+1))}
Showing 1-5 of 5 results.