A385319
a(n) = Sum_{k=0..n} 2^k * binomial(2*n,k) * binomial(2*n-k-1,n-k).
Original entry on oeis.org
1, 5, 43, 422, 4387, 47090, 515854, 5731052, 64330531, 727812026, 8285505178, 94798502804, 1089146648206, 12556967516852, 145201851788092, 1683334752235352, 19558532125813027, 227694254392461962, 2655343386035416162, 31014205667706302852, 362746369474101224602
Offset: 0
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Table[Sum[3^k*(-1)^(n-k)*Binomial[2*n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 31 2025 *)
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a(n) = sum(k=0, n, 2^k*binomial(2*n,k)*binomial(2*n-k-1, n-k));
A385320
a(n) = Sum_{k=0..n} 2^k * binomial(3*n,k) * binomial(3*n-k-1,n-k).
Original entry on oeis.org
1, 8, 118, 1970, 34714, 630548, 11678284, 219240008, 4157096266, 79429466456, 1526869550638, 29495424821354, 572100064904872, 11134578632483600, 217341014671302976, 4253067310380772400, 83409477100625759050, 1638952453699219007072, 32259670449587082804466
Offset: 0
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Table[Sum[3^k*(-1)^(n-k)*Binomial[3*n, k], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jul 31 2025 *)
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a(n) = sum(k=0, n, 2^k*binomial(3*n, k)*binomial(3*n-k-1, n-k));
A386723
Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1+2*x)^4 ).
Original entry on oeis.org
1, 11, 175, 3275, 67156, 1460237, 33073930, 771961835, 18437940220, 448483875596, 11071403236807, 276675755470349, 6985664542196380, 177932236341440270, 4566561255466298500, 117974930924420353835, 3065563791639454312492, 80069021664742889373380
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3/(1+2*x)^4)/x)
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a(n) = sum(k=0, n, 2^k*binomial(4*(n+1), k)*binomial(4*n-k+2, n-k))/(n+1);
A384366
a(n) = Sum_{k=0..n} 2^k * binomial(4*n+1,k) * binomial(4*n-k,n-k).
Original entry on oeis.org
1, 14, 298, 7058, 175594, 4494104, 117160486, 3094165004, 82503894826, 2216251440200, 59884814271208, 1625891941764962, 44318988449261926, 1212105802241702408, 33245450748860850532, 914105822029066709048, 25188189341369313927082, 695379304005363364395752
Offset: 0
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a(n) = sum(k=0, n, 2^k*binomial(4*n+1, k)*binomial(4*n-k, n-k));
Showing 1-4 of 4 results.