A377148
a(n) = Sum_{k=0..n} binomial(k+3,3) * binomial(k,n-k)^2.
Original entry on oeis.org
1, 4, 14, 60, 225, 796, 2764, 9304, 30580, 98700, 313422, 981548, 3037473, 9301620, 28222000, 84927760, 253699285, 752863840, 2220831160, 6515581600, 19021079866, 55276625304, 159967084164, 461150383400, 1324652146775, 3792447499916, 10824189204014
Offset: 0
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[&+[Binomial(k+3,3)*Binomial(k, n-k)^2: k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, May 12 2025
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Table[Sum[Binomial[k+3,3]*Binomial[k, n-k]^2,{k,0,n}],{n,0,30}] (* Vincenzo Librandi, May 12 2025 *)
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a(n) = sum(k=0, n, binomial(k+3, 3)*binomial(k, n-k)^2);
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a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=3, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))
A377145
a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(k,n-k)^2.
Original entry on oeis.org
1, 3, 9, 34, 111, 351, 1103, 3384, 10224, 30536, 90222, 264186, 767663, 2215623, 6356907, 18143300, 51540885, 145801395, 410888595, 1153964520, 3230723826, 9019081038, 25112021154, 69750583164, 193303849531, 534602071341, 1475644537323, 4065845732794
Offset: 0
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a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(k, n-k)^2);
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a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=2, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))
A382474
a(n) = Sum_{k=0..n} binomial(k+7,7) * binomial(2*k,2*n-2*k).
Original entry on oeis.org
1, 8, 44, 336, 2166, 11832, 60576, 292248, 1334817, 5840296, 24637976, 100684376, 400255050, 1553016960, 5897388492, 21967711160, 80425346844, 289868771928, 1029979010972, 3612517052608, 12520285820362, 42919328903928, 145643017892472, 489606988741128
Offset: 0
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
- Index entries for linear recurrences with constant coefficients, signature (16, -104, 352, -708, 1232, -2800, 5168, -6122, 9728, -18008, 16816, -19152, 37744, -27096, 24960, -50611, 24960, -27096, 37744, -19152, 16816, -18008, 9728, -6122, 5168, -2800, 1232, -708, 352, -104, 16, -1).
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[&+[Binomial(k+7, 7) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 22 2025
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Table[Sum[Binomial[k+7,7]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 22 2025 *)
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a(n) = sum(k=0, n, binomial(k+7, 7)*binomial(2*k, 2*n-2*k));
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my(N=7, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))
A377152
a(n) = Sum_{k=0..n} binomial(k+4,4) * binomial(k,n-k)^2.
Original entry on oeis.org
1, 5, 20, 95, 400, 1561, 5915, 21610, 76585, 265075, 898622, 2992235, 9810290, 31727815, 101379175, 320464280, 1003259080, 3113576320, 9586763720, 29305985800, 88997753446, 268642069750, 806394498200, 2408144329250, 7157177344225, 21177323087891
Offset: 0
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f:= proc(n) local k; add(binomial(k+4,4)*binomial(k,n-k)^2,k=0..n) end proc:
map(f, [$0..50]); # Robert Israel, Dec 05 2024
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a(n) = sum(k=0, n, binomial(k+4, 4)*binomial(k, n-k)^2);
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a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=4, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))
A377153
a(n) = Sum_{k=0..n} binomial(k+5,5) * binomial(k,n-k)^2.
Original entry on oeis.org
1, 6, 27, 140, 651, 2772, 11354, 44640, 169371, 624742, 2248575, 7922124, 27397937, 93214632, 312559200, 1034507696, 3384194616, 10954244952, 35118346760, 111602517096, 351819819414, 1100912299156, 3421515852834, 10566654790176, 32441857824859, 99060134392422
Offset: 0
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a(n) = sum(k=0, n, binomial(k+5, 5)*binomial(k, n-k)^2);
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a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=5, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))
A377158
a(n) = Sum_{k=0..n} binomial(k+6,6) * binomial(k,n-k)^2.
Original entry on oeis.org
1, 7, 35, 196, 994, 4578, 20118, 84540, 341397, 1335103, 5078227, 18852428, 68519920, 244413820, 857393700, 2963013816, 10102413972, 34025396580, 113329367816, 373642488044, 1220412680410, 3951964394642, 12695738508950, 40484919514284, 128216539026261
Offset: 0
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a(n) = sum(k=0, n, binomial(k+6, 6)*binomial(k, n-k)^2);
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a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=6, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))
Showing 1-6 of 6 results.