A336873
a(n) = Sum_{k=0..n} (binomial(n+k,k) * binomial(n,k))^n.
Original entry on oeis.org
1, 3, 73, 36729, 473940001, 155741521320033, 1453730786373283012225, 415588116056535702096428038017, 3278068950996636050857475073848209555969, 756475486389705843580676191270930552553654909184513, 5850304627708628483969594929628923064185219454493588333628772353
Offset: 0
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[(&+[(Binomial(2*j,j)*Binomial(n+j,n-j))^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 31 2022
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a[n_] := Sum[(Binomial[n+k, k] * Binomial[n, k])^n, {k, 0, n} ]; Array[a, 11, 0] (* Amiram Eldar, Aug 06 2020 *)
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{a(n) = sum(k=0, n, (binomial(n+k,k)*binomial(n,k))^n)}
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def A336873(n): return sum((binomial(2*j,j)*binomial(n+j, n-j))^n for j in (0..n))
[A336873(n) for n in (0..20)] # G. C. Greubel, Aug 31 2022
A345041
a(n) = Sum_{k=0..n} Stirling2(n,k)^n.
Original entry on oeis.org
1, 1, 2, 29, 3699, 10625002, 607758784933, 868305359018619811, 72322260589630363186583012, 141134946941935843819745493472571577, 21506852953850913182859127590586670415329232127, 213131394708948856925732826175269041102801068792839463406106
Offset: 0
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[(&+[StirlingSecond(n,j)^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 31 2022
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Table[Sum[StirlingS2[n, k]^n, {k, 0, n}], {n, 0, 11}]
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a(n) = sum(k=0, n, stirling(n, k, 2)^n) \\ Felix Fröhlich, Jun 06 2021
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def A345041(n): return sum(stirling_number2(n,j)^n for j in (0..n))
[A345041(n) for n in (0..20)] # G. C. Greubel, Aug 31 2022
A336622
a(n) = Sum_{k=0..n} Sum_{i=0..k} Sum_{j=0..i} (binomial(n,k) * binomial(k,i) * binomial(i,j))^n.
Original entry on oeis.org
1, 4, 28, 1192, 591460, 3441637504, 219272057247376, 185528149944660881488, 2405748000504972140803769860, 349789137657321307953339196885516144, 652520795984468974632890750361094911319873648
Offset: 0
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B:=Binomial; [(&+[(&+[(&+[(B(n,j)*B(n-j,k-j)*B(k-j,k-i))^n: j in [0..i]]): i in [0..k]]): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 31 2022
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Table[Sum[Sum[Sum[(Binomial[n, k] Binomial[k, i] Binomial[i, j])^n, {j, 0, i}], {i, 0, k}], {k, 0, n}], {n, 0, 10}]
Table[(n!)^n SeriesCoefficient[Sum[x^k/(k!)^n, {k, 0, n}]^4, {x, 0, n}], {n, 0, 10}]
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b=binomial
def A336622(n): return sum(sum(sum( (b(n,j)*b(n-j,k-j)*b(k-j,k-i))^n for j in (0..i)) for i in (0..k)) for k in (0..n))
[A336622(n) for n in (0..20)] # G. C. Greubel, Aug 31 2022
