Search: a323206 -id:a323206
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A323208
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a(n) = hypergeometric([-n - 1, n + 2], [-n - 2], n).
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+10
3
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1, 5, 67, 1606, 55797, 2537781, 142648495, 9549411950, 741894295369, 65620725560578, 6511108452179611, 716273662860469000, 86527644431076024637, 11387523335268377432565, 1621766490238904658104583, 248507974510512755641561366, 40769019250019155227631614225
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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) = Sum_{j=0..n+1} (binomial(2*(n+1)-j,n+1)-binomial(2*(n+1)-j,n+2))*n^(n+1-j).
a(n) = Sum_{j=0..n+1} binomial(n+1+j, n+1)*(1 - j/(n+2))*n^j.
a(n) = 1 + Sum_{j=0..n} ((1+j)*binomial(2*(n+1)-j, n+2)/(n+1-j))*n^(n+1-j).
a(n) = (1/(2*Pi))*Integral_{x=0..4*n} (sqrt(x*(4*n-x))*x^(n+1))/(1+(n-1)*x), n>0.
a(n) ~ (4^(n + 2)*n^(n + 3))/(sqrt(Pi)*(1 - 2*n)^2*(n + 1)^(3/2)).
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MAPLE
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# The function ballot is defined in A238762.
a := n -> add(ballot(2*j, 2*n+2)*n^j, j=0..n+1):
seq(a(n), n=0..16);
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MATHEMATICA
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a[n_] := Hypergeometric2F1[-n - 1, n + 2, -n - 2, n];
Table[a[n], {n, 0, 16}]
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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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A323207
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a(n) = Sum_{k=0..n} hypergeometric([-k, k + 1], [-k - 1], n - k).
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+10
2
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1, 2, 4, 10, 33, 141, 752, 4825, 36027, 305132, 2879840, 29909421, 338479429, 4139716658, 54339861530, 761150445734, 11322139144239, 178116143657889, 2952831190016238, 51423702126549166, 938126972940647197, 17883424301972473339
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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) = Sum_{k=0..n} A323206(n-k, k).
a(n) = Sum_{k=0..n} Sum_{j=0..k} A238762(2*j, 2*k)*(n-k)^j.
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} (binomial(2*(n-k)-j, n-k) - binomial(2*(n-k)-j, n-k+1))*k^(n-k-j).
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MAPLE
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# The function ballot is defined in A238762.
A323207 := n -> add(add(ballot(2*j, 2*k)*(n-k)^j, j=0..k), k=0..n):
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MATHEMATICA
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a[n_] := Sum[Hypergeometric2F1[-k, k + 1, -k - 1, n - k], {k, 0, n}];
Table[a[n], {n, 0, 21}]
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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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A323217
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a(n) = hypergeometric([-n, n + 1], [-n - 1], n + 1).
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+10
2
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1, 3, 25, 413, 10746, 387607, 17981769, 1022586105, 68964092542, 5384626548491, 477951767068986, 47546350648784341, 5240644323742274500, 634033030117301108127, 83540992651137240168361, 11908866726507685451458545
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graph;
refs;
listen;
history;
text;
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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) = Sum_{j=0..n} (binomial(2*n-j, n) - binomial(2*n-j, n+1))*(n+1)^(n-j).
a(n) = Sum_{j=0..n} binomial(n+j, n)*(1 - j/(n + 1))*(n + 1)^j.
a(n) = 1 + Sum_{j=0..n-1} ((1+j)*binomial(2*n-j, n+1)/(n-j))*(n+1)^(n-j).
a(n) = (1/(2*Pi))*Integral_{x=0..4*(n+1)} (sqrt(x*(4*(n+1)-x))*x^n)/(1+n*x).
a(n) ~ (4^(n+1)*(n+1)^(n+2))/(sqrt(Pi)*(2*n+1)^2*n^(3/2)).
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MAPLE
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# The function ballot is defined in A238762.
a := n -> add(ballot(2*j, 2*n)*(n+1)^j, j=0..n):
seq(a(n), n=0..16);
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MATHEMATICA
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a[n_] := Hypergeometric2F1[-n, n + 1, -n - 1, n + 1];
Table[a[n], {n, 0, 16}]
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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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A323209
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a(n) = hypergeometric([-n, n + 1], [-n - 1], n).
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+10
1
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1, 2, 13, 190, 4641, 161376, 7312789, 409186310, 27272680705, 2110472708140, 186023930383501, 18401769878685172, 2018938571514794593, 243319689384354960300, 31955654188732155634341, 4542582850906442990797126, 694922224386422689648830465
(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) = Sum_{j=0..n} (binomial(2*n-j, n) - binomial(2*n-j, n+1))*n^(n-j).
a(n) = Sum_{j=0..n} binomial(n+j, n)*(1 - j/(n + 1))*n^j.
a(n) = 1 + Sum_{j=0..n-1} ((1+j)*binomial(2*n-j, n+1)/(n-j))*n^(n-j).
a(n) = (1/(2*Pi))*Integral_{x=0..4*n} (sqrt(x*(4*n-x))*x^n)/(1+(n-1)*x), n>0.
a(n) ~ (4^(n + 1)*n^(n + 1/2))/(sqrt(Pi)*(1 - 2*n)^2).
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MAPLE
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# The function ballot is defined in A238762.
a := n -> add(ballot(2*k, 2*n)*n^k, k=0..n):
seq(a(n), n=0..16);
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MATHEMATICA
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a[n_] := Hypergeometric2F1[-n, n + 1, -n - 1, n];
Table[a[n], {n, 0, 14}]
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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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