A132688
a(n) = binomial(2^n + 3*n, n).
Original entry on oeis.org
1, 5, 45, 680, 20475, 1533939, 350161812, 280384608504, 847073824772175, 9894081531608130857, 446730013630787463700695, 77328499046923986969058944720, 50891283683781760304442885961988100
Offset: 0
Sequences of the form binomial(2^n +p*n +q, n):
A136556 (0,-1),
A014070 (0,0),
A136505 (0,1),
A136506 (0,2),
A060690 (1,-1),
A132683 (1,0),
A132684 (1,1),
A132685 (2,0),
A132686 (2,1),
A132687 (3,-1), this sequence (3,0),
A132689 (3,1).
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[Binomial(2^n +3*n, n): n in [0..20]]; // G. C. Greubel, Mar 13 2021
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Table[Binomial[2^n+3n,n],{n,0,20}] (* Harvey P. Dale, Oct 30 2018 *)
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a(n)=binomial(2^n+3*n,n)
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[binomial(2^n +3*n, n) for n in (0..20)] # G. C. Greubel, Mar 13 2021
A132689
a(n) = binomial(2^n + 3*n + 1, n).
Original entry on oeis.org
1, 6, 55, 816, 23751, 1712304, 377447148, 294109729200, 871896500955975, 10061777828754031380, 451004941990890693018405, 77739225019650285306412710240, 51039474754930845750609669420261300
Offset: 0
Sequences of the form binomial(2^n +p*n +q, n):
A136556 (0,-1),
A014070 (0,0),
A136505 (0,1),
A136506 (0,2),
A060690 (1,-1),
A132683 (1,0),
A132684 (1,1),
A132685 (2,0),
A132686 (2,1),
A132687 (3,-1),
A132688 (3,0), this sequence (3,1).
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[Binomial(2^n+3*n+1, n) : n in [0..15]]; // Wesley Ivan Hurt, Nov 20 2014
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A132689:=n->binomial(2^n+3*n+1, n): seq(A132689(n), n=0..15); # Wesley Ivan Hurt, Nov 20 2014
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Table[Binomial[2^n +3n +1, n], {n, 0, 15}] (* Wesley Ivan Hurt, Nov 20 2014 *)
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a(n)=binomial(2^n+3*n+1,n)
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[binomial(2^n +3*n+1, n) for n in (0..15)] # G. C. Greubel, Feb 15 2021
A136507
a(n) = Sum_{k=0..n} binomial(2^(n-k) + k, n-k).
Original entry on oeis.org
1, 3, 10, 71, 1925, 203904, 75214965, 94608676477, 409763735870986, 6208539881584781823, 334272186911271376874561, 64832512634295914941490910360, 45811927207957062190019240099653265
Offset: 0
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[(&+[Binomial(2^k +n-k, k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Mar 14 2021
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A136507:= n-> add(binomial(2^k +n-k, k), k=0..n); seq(A136507(n), n=0..20); # G. C. Greubel, Mar 14 2021
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Table[Sum[Binomial[2^(n-k)+k,n-k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Mar 08 2015 *)
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{a(n)=sum(k=0,n,binomial(2^(n-k)+k,n-k))}
for(n=0,16, print1(a(n),", "))
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/* a(n) = coefficient of x^n in o.g.f. series: */
{a(n)=polcoeff(sum(i=0,n,1/(1-x-2^i*x^2 +x*O(x^n))*log(1+2^i*x +x*O(x^n))^i/i!),n)}
for(n=0,16, print1(a(n),", "))
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[sum(binomial(2^k +n-k, k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Mar 14 2021