A335309
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * n^(n-k).
Original entry on oeis.org
1, 3, 22, 245, 3606, 65527, 1411404, 35066313, 985483270, 30869546411, 1065442493556, 40144438269949, 1638733865336764, 72012798200637855, 3388250516614331416, 169894851136173584145, 9041936334960057699654, 508945841697238471315027, 30202327515992972576218980
Offset: 0
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Join[{1}, Table[Sum[Binomial[n, k] Binomial[n + k, k] n^(n - k), {k, 0, n}], {n, 1, 18}]]
Table[SeriesCoefficient[1/Sqrt[1 - 2 (n + 2) x + n^2 x^2], {x, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[Exp[(n + 2) x] BesselI[0, 2 Sqrt[n + 1] x], {x, 0, n}], {n, 0, 18}]
Table[Hypergeometric2F1[-n, -n, 1, 1 + n], {n, 0, 18}]
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a(n) = sum(k=0, n, binomial(n,k)^2*(n+1)^k); \\ Michel Marcus, Jun 01 2020
A367256
a(n) = Sum_{k=0..n} binomial(n, k) * binomial(n - 1, k - 1) * n^(n - k).
Original entry on oeis.org
1, 1, 5, 46, 593, 9726, 192637, 4457580, 117769409, 3492894070, 114790042901, 4137157889316, 162154385331985, 6863637142316332, 311905306734621069, 15140756439172826776, 781693659313991730945, 42759819036520142319270, 2469943332976774829606821
Offset: 0
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a := n -> if n= 0 then 1 else n*n^(n - 1)*hypergeom([1 - n, 1 - n], [2], 1/n) fi:
seq(simplify(a(n)), n = 0..19);
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A367256[n_] := If[n == 0, 1, n*n^(n-1)*Hypergeometric2F1[1-n, 1-n, 2, 1/n]];
Array[A367256, 25, 0] (* Paolo Xausa, Jan 31 2024 *)
A188387
Central coefficient in (1 + (2^n+1)*x + 2^n*x^2)^n for n>=0.
Original entry on oeis.org
1, 3, 33, 1161, 140545, 63148833, 111254837505, 793938286762113, 23282575640347295745, 2812444483776375381074433, 1393909730376211388561041231873
Offset: 0
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/*1*/ P:=PolynomialRing(Integers()); [ Coefficients((1+(2^n+1)*x+2^n*x^2)^n)[n+1]: n in [0..10] ]; /*2*/ &cat[ [&+[ Binomial(n, k)^2*2^(n*k): k in [0..n]]]: n in [0..10] ]; // Bruno Berselli, Mar 30 2011
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Table[Sum[Binomial[n,k]^2 * 2^(n*k), {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Feb 11 2015 *)
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{a(n)=polcoeff((1+(2^n+1)*x+2^n*x^2+x*O(x^n))^n,n)}
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{a(n)=sum(k=0,n,binomial(n,k)^2*2^(n*k))}
A383132
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n*k,k) * n^k.
Original entry on oeis.org
1, 2, 33, 2701, 524993, 181752001, 97735073905, 75179269556672, 78240951854025217, 105806762566689176353, 180297512864534759056001, 377878889913778527874694227, 955217573424445946022789385537, 2865620569274978738097814056365899, 10064763360358683666070320479027168465
Offset: 0
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Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k] Binomial[n k, k] n^k, {k, 0, n}], {n, 0, 14}]