A345455
a(n) = Sum_{k=0..n} binomial(5*n+1,5*k).
Original entry on oeis.org
1, 7, 474, 12393, 427351, 13333932, 430470899, 13733091643, 439924466026, 14072420067757, 450374698997499, 14411355379952868, 461170414282959151, 14757375158697584607, 472236871202375365274, 15111570273013075344193, 483570355262634763462351
Offset: 0
Sum_{k=0..n} binomial(b*n+c,b*k):
A082311 (b=3,c=1),
A090407 (b=4,c=1),
A070782 (b=5,c=0), this sequence (b=5,c=1),
A345456 (b=5,c=2),
A345457 (b=5,c=3),
A345458 (b=5,c=4).
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a[n_] := Sum[Binomial[5*n + 1, 5*k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Jun 20 2021 *)
LinearRecurrence[{21,353,-32},{1,7,474},20] (* Harvey P. Dale, Jul 20 2021 *)
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a(n) = sum(k=0, n, binomial(5*n+1, 5*k));
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my(N=20, x='x+O('x^N)); Vec((1-14*x-26*x^2)/((1-32*x)*(1+11*x-x^2)))
A345456
a(n) = Sum_{k=0..n} binomial(5*n+2,5*k).
Original entry on oeis.org
1, 22, 859, 25773, 843756, 26789257, 859595529, 27481113638, 879683351911, 28146676447417, 900729032983924, 28822936611339453, 922338323835136341, 29514778095285204502, 944473434343229560419, 30223143962480773595093, 967140672636207153780796
Offset: 0
Sum_{k=0..n} binomial(b*n+c,b*k):
A070782 (b=5,c=0),
A345455 (b=5,c=1), this sequence (b=5,c=2),
A345457 (b=5,c=3),
A345458 (b=5,c=4).
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a[n_] := Sum[Binomial[5*n + 2, 5*k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Jun 20 2021 *)
LinearRecurrence[{21,353,-32},{1,22,859},20] (* Harvey P. Dale, Aug 25 2022 *)
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a(n) = sum(k=0, n, binomial(5*n+2, 5*k));
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my(N=20, x='x+O('x^N)); Vec((1+x+44*x^2)/((1-32*x)*(1+11*x-x^2)))
A345458
a(n) = Sum_{k=0..n} binomial(5*n+4,5*k).
Original entry on oeis.org
1, 127, 3004, 107883, 3321891, 107746282, 3431847189, 109996928003, 3517929664756, 112595619434887, 3602817278095399, 115292842751246298, 3689341137121931721, 118059247217851456567, 3777892242010882603884, 120892592433742197034643
Offset: 0
Sum_{k=0..n} binomial(b*n+c,b*k):
A070782 (b=5,c=0),
A345455 (b=5,c=1),
A345456 (b=5,c=2),
A345457 (b=5,c=3), this sequence (b=5,c=4).
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a[n_] := Sum[Binomial[5*n + 4, 5*k], {k, 0, n}]; Array[a, 16, 0] (* Amiram Eldar, Jun 20 2021 *)
Total/@Table[Binomial[5n+4,5k],{n,0,20},{k,0,n}] (* or *) LinearRecurrence[{21,353,-32},{1,127,3004},30] (* Harvey P. Dale, Oct 29 2023 *)
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a(n) = sum(k=0, n, binomial(5*n+4, 5*k));
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my(N=20, x='x+O('x^N)); Vec((1+106*x-16*x^2)/((1-32*x)*(1+11*x-x^2)))
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