A262441
a(n) = Sum_{k=0..n+1}(binomial(n-1,k)/(k+1)*binomial(n+k+1,n-k)).
Original entry on oeis.org
1, 2, 5, 16, 58, 226, 924, 3910, 16979, 75232, 338776, 1545886, 7132580, 33219086, 155963851, 737383488, 3507680650, 16776206680, 80622416976, 389123999656, 1885405316596, 9167409871040, 44717351734160, 218762640192838, 1073082055680180
Offset: 0
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[&+[Binomial(n-1, k)/(k+1)*Binomial(n+k+1, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 23 2015
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Join[{1}, Table[Sum[ Binomial[n-1, k] / (k+1) Binomial[ n+k+1, n-k], {k, 0, n+1}], {n, 25}]] (* Vincenzo Librandi, Sep 23 2015 *)
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a(n):=sum(binomial(n,k)*binomial(n+k-2,n-k-1),k,0,n-1)/n;
A(x):=sum(a(n)*x^n,n,1,30);
taylor((1/x-1/A(x)),x,0,10);
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a(n)=sum(k=0,n+1,(binomial(n-1,k)/(k+1)*binomial(n+k+1,n-k))) \\ Anders Hellström, Sep 23 2015
A161799
G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^2)^3.
Original entry on oeis.org
1, 3, 12, 61, 345, 2085, 13182, 86106, 576543, 3936029, 27294390, 191722887, 1361291244, 9754412169, 70447946556, 512278417176, 3747570671685, 27561220671408, 203657352324178, 1511270129552163, 11257532921742528
Offset: 0
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A161799 := proc(n)
local s,t ;
s := 2 ;
t := 3;
add( binomial(t*n-(t-1)*(k-1),k) * binomial(n+(s-1)*k-1,n-k) /(n-k+1) ,k=0..n) ;
end proc:
seq(A161799(n),n=0..40) ; # R. J. Mathar, May 12 2022
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Table[Sum[Binomial[3*n-2*k+2,k]/(n-k+1)*Binomial[n+k-1,n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Sep 18 2013 *)
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{a(n,m=1)=sum(k=0,n,binomial(3*n-2*k+3*m-1,k)*m/(n-k+m)*binomial(n+k-1,n-k))}
A349022
G.f. satisfies A(x) = 1/(1 - x/(1 - x*A(x))^3)^4.
Original entry on oeis.org
1, 4, 22, 152, 1161, 9460, 80550, 708172, 6379368, 58576168, 546215580, 5158542152, 49239812893, 474285453628, 4604149947276, 44999181550032, 442430807369519, 4372944634271688, 43425156714959956, 433049078716727332, 4334925824762251939
Offset: 0
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A349022 := proc(n)
add(binomial(4*n-3*(k-1),k)*binomial(n+2*k-1,n-k)/(n-k+1),k=0..n) ;
end proc:
seq(A349022(n),n=0..40) ; # R. J. Mathar, Jan 25 2023
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a(n, s=3, t=4) = sum(k=0, n, binomial(t*n-(t-1)*(k-1), k)*binomial(n+(s-1)*k-1, n-k)/(n-k+1));
A350290
a(n) = Sum_{k=0..n} (-1)^(n - k) * binomial(n, k) * binomial(n + k - 1, n - k).
Original entry on oeis.org
1, 1, -3, -2, 21, -4, -150, 155, 1029, -2072, -6468, 22056, 34122, -208857, -106249, 1816958, -639067, -14629264, 17635800, 108117620, -239571684, -711876496, 2628772968, 3825823888, -25582846134, -10997156129, 227594431035, -98360217830, -1864646227185
Offset: 0
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a := n -> add((-1)^(n - k)*binomial(n, k)*binomial(n + k - 1, n-k), k = 0..n):
seq(a(n), n = 0..28);
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a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*binomial(n+k-1, n-k)); \\ Michel Marcus, Mar 07 2022
Showing 1-4 of 4 results.