A378464
a(n) = Sum_{k=0..floor(n/3)} binomial(n+k-1,k) * binomial(2*n-1,n-3*k).
Original entry on oeis.org
1, 1, 3, 13, 63, 306, 1473, 7085, 34239, 166459, 813618, 3994200, 19678233, 97239130, 481740885, 2392004853, 11900655999, 59312062026, 296071376307, 1479998924447, 7407613846698, 37118966710076, 186195636158436, 934889598483048, 4698229684691913, 23629859054461331
Offset: 0
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a(n) = sum(k=0, n\3, binomial(n+k-1, k)*binomial(2*n-1, n-3*k));
A383479
Number of lattice paths from (0,0) to (n,n) using steps (1,0),(3,0),(0,1).
Original entry on oeis.org
1, 2, 6, 24, 100, 420, 1792, 7752, 33858, 148940, 658944, 2929056, 13070876, 58521344, 262754040, 1182619280, 5334172518, 24104916504, 109111142376, 494630028200, 2245300152480, 10204575481320, 46429481139000, 211460450151600, 963971663881200, 4398118872144192
Offset: 0
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f:= proc(x,y) option remember;
local t;
t:= 0;
if x >= 1 then t:= t + procname(x-1,y) fi;
if x >= 3 then t:= t + procname(x-3,y) fi;
if y >= 1 then t:= t + procname(x,y-1) fi;
t
end proc:
f(0,0):= 1:
seq(f(n,n),n=0..25); # Robert Israel, May 28 2025
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a(n) = sum(k=0, n\3, binomial(n+k, k)*binomial(2*n-2*k, n-3*k));
A383481
Coefficient of x^n in the expansion of 1 / (1-x-x^4)^n.
Original entry on oeis.org
1, 1, 3, 10, 39, 156, 630, 2556, 10431, 42823, 176748, 732810, 3049722, 12732188, 53299284, 223645200, 940355391, 3961092906, 16712516565, 70615352330, 298761296064, 1265504676810, 5366250376710, 22777466596560, 96768003904650, 411451657313931, 1750809473690436, 7455339422353396
Offset: 0
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f:= proc(n) local k; add(binomial(n+k-1,k)*binomial(2*n-3*k-1,n-4*k),k=0..n/4) end proc:
map(f, [$0..40]); # Robert Israel, May 28 2025
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a(n, s=4, t=1, u=0) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t-u+1)*n-(s-1)*k-1, n-s*k));
A378463
a(n) = Sum_{k=0..floor(n/3)} binomial(n+k-1,k) * binomial(2*n-k-1,n-3*k).
Original entry on oeis.org
1, 1, 3, 13, 59, 266, 1203, 5489, 25259, 117022, 545038, 2549592, 11970035, 56372460, 266194295, 1259910113, 5975382699, 28390616727, 135108035502, 643891031826, 3072604703774, 14679493913048, 70206875750168, 336103001918788, 1610476039036259, 7723148579525441
Offset: 0
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a(n) = sum(k=0, n\3, binomial(n+k-1, k)*binomial(2*n-k-1, n-3*k));
Showing 1-4 of 4 results.