A387338
a(n) = Sum_{k=0..n} 2^k * binomial(n+3,k) * binomial(n+3,k+3).
Original entry on oeis.org
1, 12, 100, 720, 4809, 30744, 191184, 1167120, 7033785, 41999364, 249075684, 1469561184, 8636441905, 50600529840, 295755641152, 1725379046496, 10050215851665, 58470232877820, 339832224226180, 1973538115293360, 11453616812552761, 66436765880135112
Offset: 0
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[&+[2^k * Binomial(n+3,k) * Binomial(n+3,k+3): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025
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Table[Sum[2^k * Binomial[n+3,k]*Binomial[n+3, k+3],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *)
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a(n) = sum(k=0, n, 2^k*binomial(n+3, k)*binomial(n+3, k+3));
A387340
a(n) = Sum_{k=0..n} 3^k * binomial(n+3,k) * binomial(n+3,k+3).
Original entry on oeis.org
1, 16, 175, 1640, 14189, 117152, 939036, 7379040, 57188010, 438810592, 3342302821, 25316084248, 190937278805, 1435287936320, 10760879892008, 80509920297792, 601343784616830, 4485466826475360, 33420579148668670, 248788060638391120, 1850652536242372786
Offset: 0
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[&+[3^k * Binomial(n+3,k) * Binomial(n+3,k+3): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025
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Table[Sum[3^k * Binomial[n+3,k]*Binomial[n+3, k+3],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *)
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a(n) = sum(k=0, n, 3^k*binomial(n+3, k)*binomial(n+3, k+3));
A126219
Triangle read by rows: T(n,k) is the number of binary trees (i.e., a rooted tree where each vertex has either 0, 1, or 2 children; and, when only one child is present, it is either a right child or a left child) with n edges and k pairs of adjacent vertices of outdegree 2.
Original entry on oeis.org
1, 2, 5, 14, 40, 2, 116, 16, 344, 80, 5, 1040, 340, 50, 3188, 1360, 300, 14, 9880, 5264, 1484, 168, 30912, 19880, 6776, 1176, 42, 97520, 73728, 29568, 6608, 588, 309856, 269952, 124656, 33600, 4704, 132, 990656, 979264, 511584, 161280, 29544, 2112
Offset: 0
Triangle starts:
1;
2;
5;
14;
40, 2;
116, 16;
344, 80, 5;
1040, 340, 50;
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G:=1/2*(1-4*z^3*t^2-4*z^3-2*z^2*t+8*z^3*t-2*z+2*z^2*t^2-sqrt(1-8*z^3+4*z^2-4*z^2*t-4*z+8*z^3*t))/z^2/(2*z*t-t-2*z)^2: Gser:=simplify(series(G,z=0,18)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: 1;2; for n from 2 to 14 do seq(coeff(P[n],t,j),j=0..floor(n/2)-1) od; # yields sequence in triangular form
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