A360100
a(n) = Sum_{k=0..n} binomial(n+2*k-1,n-k) * Catalan(k).
Original entry on oeis.org
1, 1, 5, 23, 111, 562, 2952, 15948, 88076, 495077, 2823293, 16295020, 95007654, 558765743, 3310999269, 19748462718, 118471172054, 714355994997, 4327148812557, 26319195869861, 160677354596769, 984236344800234, 6047526697800992, 37262944840704171
Offset: 0
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A360100 := proc(n)
add(binomial(n+2*k-1,n-k)*A000108(k),k=0..n) ;
end proc:
seq(A360100(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
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m = 24;
A[_] = 0;
Do[A[x_] = 1 + x A[x]^2/(1 - x)^3 + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Aug 16 2023 *)
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a(n) = sum(k=0, n, binomial(n+2*k-1, n-k)*binomial(2*k, k)/(k+1));
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my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x/(1-x)^3)))
A258973
The number of plain lambda terms presented by de Bruijn indices, see Bendkowski et al., where zeros have no weight.
Original entry on oeis.org
1, 3, 10, 40, 181, 884, 4539, 24142, 131821, 734577, 4160626, 23881695, 138610418, 812104884, 4796598619, 28529555072, 170733683579, 1027293807083, 6211002743144, 37713907549066, 229894166951757, 1406310771154682, 8630254073158599, 53117142215866687, 327800429456036588
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, and Marek Zaionc, Combinatorics of λ-terms: a natural approach, arXiv:1609.08106 [cs.LO], 2016.
- Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, and Marek Zaionc, A Natural Counting of Lambda Terms, arXiv preprint arXiv:1506.02367 [cs.LO], 2015.
- Maciej Bendkowski and Pierre Lescanne, On the enumeration of closures and environments with an application to random generation, Logical Methods in Computer Science (2019) Vol. 15, No. 4, 3:1-3:21.
- K. Grygiel and P. Lescanne, A natural counting of lambda terms, Preprint 2015.
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a:= proc(n) option remember; `if`(n<4, [1, 3, 10, 40][n+1],
((8*n-3)*a(n-1)-(10*n-13)*a(n-2)
+(4*n-11)*a(n-3)-(n-4)*a(n-4))/(n+1))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Jun 30 2015
a := n -> add(hypergeom([(i+1)/2, i/2+1, i-n+1], [1, 2], -4), i=0..n-1):
seq(simplify(a(n)), n=0..25); # Peter Luschny, May 03 2018
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a[n_] := a[n] = If[n<4, {1, 3, 10, 40}[[n+1]], ((8*n-3)*a[n-1] - (10*n-13)*a[n-2] + (4*n-11)*a[n-3] - (n-4)*a[n-4])/(n+1)]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 22 2015, after Alois P. Heinz *)
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a(n):=sum(sum((binomial(k+i-1,k-1)*binomial(2*k+i-2,k+i-1)*binomial(n-i-1,n-k-i))/k,k,1,n-i),i,0,n); /* Vladimir Kruchinin, May 03 2018 */
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lista(nn) = {z = y + O(y^nn); Vec(((1-z)^2 - sqrt((1-z)^4-4*z*(1-z))) / (2*z*(1-z)));} \\ Michel Marcus, Jun 30 2015
A360103
a(n) = Sum_{k=0..n} binomial(n+4*k,n-k) * Catalan(k).
Original entry on oeis.org
1, 2, 9, 49, 283, 1715, 10793, 69906, 463031, 3122264, 21363065, 147951489, 1035173405, 7306326465, 51959150713, 371950057003, 2678083379707, 19381867703946, 140915907625531, 1028760981192771, 7538511404971231, 55427326349613665, 408789584900354397
Offset: 0
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A360103 := proc(n)
add(binomial(n+4*k,n-k)*A000108(k),k=0..n) ;
end proc:
seq(A360103(n),n=0..40) ; # R. J. Mathar, Mar 12 2023
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a(n) = sum(k=0, n, binomial(n+4*k, n-k)*binomial(2*k, k)/(k+1));
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my(N=30, x='x+O('x^N)); Vec(2/((1-x)*(1+sqrt(1-4*x/(1-x)^5))))
Showing 1-3 of 3 results.