Search: a370098 -id:a370098
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A370102
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a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(5*n-k-1,n-k).
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+10
4
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1, 8, 128, 2312, 44032, 864008, 17282432, 350353928, 7172939776, 147972367880, 3070951360128, 64044689834760, 1341056098444288, 28176478479561992, 593725756425591680, 12542160174109922312, 265525958014053580800, 5632170795392966388744
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = [x^n] ( (1+x)^4/(1-x)^4 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1-x)^4/(1+x)^4 ). See A365847.
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PROG
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(PARI) a(n) = sum(k=0, n, binomial(4*n, k)*binomial(5*n-k-1, n-k));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A370103
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a(n) = Sum_{k=0..n} (-1)^k * binomial(2*n+k-1,k) * binomial(4*n-k-1,n-k).
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+10
3
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1, 1, 7, 28, 151, 751, 3976, 20924, 112023, 602182, 3260257, 17724928, 96766072, 529977917, 2910984412, 16027963528, 88440034711, 488918693466, 2707393587802, 15014647096172, 83380131228401, 463593653171495, 2580426581343200, 14377474236172320
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = [x^n] 1/( (1+x)^2 * (1-x)^3 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1+x)^2*(1-x)^3 ). See A365854.
a(n) = Sum_{k=0..floor(n/2)} binomial(3*n+k-1,k) * binomial(n,n-2*k).
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+k-1,k) * binomial(2*n-2*k-1,n-2*k).
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PROG
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(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(2*n+k-1, k)*binomial(4*n-k-1, n-k));
(PARI) a(n, s=2, t=3, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial(u*n, n-s*k));
(PARI) a(n, s=2, t=2, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((u+1)*n-s*k-1, n-s*k));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A370099
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a(n) = Sum_{k=0..n} binomial(2*n,k) * binomial(3*n-k-1,n-k).
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+10
2
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1, 4, 32, 292, 2816, 28004, 284000, 2919620, 30316544, 317222212, 3339504032, 35329425124, 375282559232, 4000059761572, 42760427177696, 458259268924292, 4921911787962368, 52965710906750084, 570951048018417440, 6164049197776406180, 66639047280436354816
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = [x^n] ( (1+x)^2/(1-x)^2 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1-x)^2/(1+x)^2 ).
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PROG
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(PARI) a(n) = sum(k=0, n, binomial(2*n, k)*binomial(3*n-k-1, n-k));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A370097
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a(n) = Sum_{k=0..n} binomial(3*n,k) * binomial(3*n-k-1,n-k).
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+10
1
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1, 5, 49, 545, 6401, 77505, 956929, 11976193, 151388161, 1928363009, 24712450049, 318255628289, 4115300220929, 53396370030593, 694845537386497, 9064787191660545, 118516719269445633, 1552528215946035201, 20372392543502991361, 267736366910401413121
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = [x^n] ( (1+x)^3/(1-x)^2 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1-x)^2/(1+x)^3 ). See A365842.
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PROG
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(PARI) a(n) = sum(k=0, n, binomial(3*n, k)*binomial(3*n-k-1, n-k));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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