A382925
a(n) = [x^(3*n)] Product_{k=0..n} (1 + k*x)^4.
Original entry on oeis.org
1, 4, 248, 61320, 39194896, 51699564000, 122482878310656, 474300956527856640, 2804126507444905046272, 24036712401508315774848000, 286889291626307627568309995520, 4615084616716397442547883972818944, 97421519516367186622078306709619806208
Offset: 0
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Abs[Table[Sum[StirlingS1[n+1, i+1] * StirlingS1[n+1, j+1] * StirlingS1[n+1, k+1] * StirlingS1[n+1, n-i-j-k+1], {i, 0, n}, {j, 0, n-i}, {k, 0, n-i-j}], {n, 0, 15}]] (* Vaclav Kotesovec, May 22 2025 *)
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a(n) = sum(i=0, n, sum(j=0, n-i, sum(k=0, n-i-j, abs(stirling(n+1, i+1, 1)*stirling(n+1, j+1, 1)*stirling(n+1, k+1, 1)*stirling(n+1, n-i-j-k+1, 1)))));
A384086
a(n) = [x^n] Product_{k=1..n} ((1 + k*x)/(1 - k*x))^2.
Original entry on oeis.org
1, 4, 72, 2352, 112000, 7023540, 546991704, 50923706176, 5517464159232, 682067031126660, 94744306830613000, 14610279918692775504, 2476682373835289303424, 457771369968515293229812, 91624876032673265663215800, 19743379886572250897986694400, 4556982707091255612929249419264
Offset: 0
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Table[SeriesCoefficient[Product[(1+k*x)^2/(1-k*x)^2, {k, 1, n}], {x, 0, n}], {n, 0, 20}]
A254927
Coefficient of x^n in Product_{k=0..n} (1+k*x)^k.
Original entry on oeis.org
1, 8, 238, 15715, 1822678, 327061056, 83839010860, 29063729300694, 13090011332041111, 7428850394493811712, 5185703819680371737432, 4366227375438927437584444, 4363140133466727238167744916, 5104897162398639619205564019232, 6912594322573705179830176812524216
Offset: 1
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Table[Coefficient[Expand[Product[(1+k*x)^k,{k,0,n}]],x^n],{n,1,20}] (* or *)
p=1; Table[p=Expand[p*(1+n*x)^n]; Coefficient[p,x^n],{n,1,20}] (* faster *)
A384025
a(n) = [x^(2*n)] Product_{k=0..n} (1 + k*x)^3.
Original entry on oeis.org
1, 3, 66, 3815, 424428, 77530530, 21106440064, 8021533034676, 4060456997959152, 2642189599046492000, 2149789283054191431744, 2139041823964877704864992, 2555760236856152336740829440, 3611539707805518014521602175296, 5958533262158042791156143146398464
Offset: 0
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Table[Sum[StirlingS1[n+1, i+1] * StirlingS1[n+1, j+1] * StirlingS1[n+1, n-i-j+1], {i, 0, n}, {j, 0, n-i}], {n, 0, 15}] (* Vaclav Kotesovec, May 22 2025 *)
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a(n) = sum(i=0, n, sum(j=0, n-i, abs(stirling(n+1, i+1, 1)*stirling(n+1, j+1, 1)*stirling(n+1, n-i-j+1, 1))));
A384028
a(n) = Sum_{k=0..2*n} Stirling1(2*n+1, 2*n+1-k) * Stirling1(2*n+1, k+1).
Original entry on oeis.org
1, 13, 2273, 1184153, 1251320145, 2232012515445, 6032418472347265, 23007314730623658225, 117745011140615270168865, 778780810721500176081199325, 6466413475830749109197652489569, 65861328745485785925705177696147337, 807448787241269228642562251336079833585
Offset: 0
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Table[Sum[StirlingS1[2*n+1, 2*n+1-j]*StirlingS1[2*n+1, j+1], {j, 0, 2*n}], {n, 0, 15}]