A371827
a(n) = Sum_{k=0..floor(n/3)} n^k * binomial(2*n-2*k,n-3*k).
Original entry on oeis.org
1, 2, 6, 23, 94, 392, 1680, 7387, 33110, 150905, 698996, 3287550, 15685420, 75877427, 371994692, 1847450970, 9290557158, 47291312897, 243574276884, 1268915237141, 6683909556420, 35585631836229, 191433293140656, 1040197718292138, 5707318227692796
Offset: 0
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Join[{1},Table[Sum[n^k Binomial[2n-2k,n-3k],{k,0,Floor[n/3]}],{n,30}]] (* Harvey P. Dale, Aug 10 2024 *)
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a(n) = sum(k=0, n\3, n^k*binomial(2*n-2*k, n-3*k));
A368892
a(n) = Sum_{k=0..floor(n/3)} n^(n-3*k) * binomial(n-2*k,k).
Original entry on oeis.org
1, 1, 4, 28, 264, 3200, 47521, 835569, 16974208, 391147867, 10080150040, 287244283821, 8967781893889, 304393809948904, 11160668048222588, 439582708115133751, 18509867068477014112, 829768603643818659302, 39454459640462073466945
Offset: 0
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Join[{1}, Table[n^n * HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, -n/3}, {1/2 - n/2, -n/2}, -27/(4*n^3)], {n, 1, 20}]] (* Vaclav Kotesovec, Jan 09 2024 *)
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a(n) = sum(k=0, n\3, n^(n-3*k)*binomial(n-2*k, k));
A368893
a(n) = Sum_{k=0..floor(n/3)} n^(n-2*k) * binomial(n-2*k,k).
Original entry on oeis.org
1, 1, 4, 30, 288, 3500, 51876, 908607, 18374656, 421492491, 10815040000, 306944040931, 9547373318400, 322972830958648, 11805432990665664, 463673398064821875, 19474259980847153152, 870954834559130974358, 41323803842611198131264
Offset: 0
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Join[{1}, Table[n^n * HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, -n/3}, {1/2 - n/2, -n/2}, -27/(4*n^2)], {n, 1, 20}]] (* Vaclav Kotesovec, Jan 09 2024 *)
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a(n) = sum(k=0, n\3, n^(n-2*k)*binomial(n-2*k, k));
A368895
a(n) = Sum_{k=0..floor(n/3)} (-n)^k * binomial(n-2*k,k).
Original entry on oeis.org
1, 1, 1, -2, -7, -14, 13, 113, 337, 19, -2579, -10867, -9911, 71852, 431229, 741181, -2178783, -20081708, -51012467, 58414532, 1061935641, 3651310699, 1841181, -62090909433, -279070619279, -250335322449, 3913178936941, 22877592319648, 38634162528361
Offset: 0
A368898
a(n) = Sum_{k=0..floor(n/4)} n^k * binomial(n-3*k,k).
Original entry on oeis.org
1, 1, 1, 1, 5, 11, 19, 29, 105, 298, 671, 1299, 3997, 12468, 33083, 75781, 220625, 708867, 2086183, 5412778, 15756741, 51093316, 160523859, 457283931, 1365001273, 4458076176, 14608351135, 44649287452, 137979763181, 455582050840, 1536403659211, 4953147876189
Offset: 0
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Table[HypergeometricPFQ[{1/4 - n/4, 1/2 - n/4, 3/4 - n/4, -n/4}, {1/3 - n/3, 2/3 - n/3, -n/3}, -256*n/27], {n, 0, 20}] (* Vaclav Kotesovec, Jan 09 2024 *)
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a(n) = sum(k=0, n\4, n^k*binomial(n-3*k, k));
Showing 1-5 of 5 results.