| Displaying 1-4 of 4 results found.
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1, 13, 109, 765, 4881, 29369, 169919, 956237, 5272945, 28632525, 153638211, 816715073, 4309138419, 22598433555, 117926579385, 612863125965, 3174156512865
COMMENTS
a(n) = sum{k=0..n} sum{i=k..n} binomial(n-k,n-i)*(2i+1)$
where i$ denotes the swinging factorial of i ( A056040).
MAPLE
swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
a := proc(n) local i, k; add(add(binomial(n-k, n-i)*swing(2*i+1), i=k..n), k=0..n) end:
Triangle interpolating the binomial transform of the swinging factorial ( A163865) with the swinging factorial ( A056040).
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4
1, 2, 1, 5, 3, 2, 16, 11, 8, 6, 47, 31, 20, 12, 6, 146, 99, 68, 48, 36, 30, 447, 301, 202, 134, 86, 50, 20, 1380, 933, 632, 430, 296, 210, 160, 140, 4251, 2871, 1938, 1306, 876, 580, 370, 210, 70, 13102, 8851, 5980, 4042, 2736, 1860, 1280, 910, 700, 630
COMMENTS
Triangle read by rows.
An analog to the binomial triangle of the factorials ( A076571).
FORMULA
T(n,k) = Sum_{i=k..n} binomial(n-k,n-i)*i$ where i$ denotes the swinging factorial of i ( A056040), for n >= 0, k >= 0.
EXAMPLE
Triangle begins
1;
2, 1;
5, 3, 2;
16, 11, 8, 6;
47, 31, 20, 12, 6;
146, 99, 68, 48, 36, 30;
447, 301, 202, 134, 86, 50, 20;
MAPLE
SumTria := proc(f, n, display) local m, A, j, i, T; T:=f(0);
for m from 0 by 1 to n-1 do A[m] := f(m);
for j from m by -1 to 1 do A[j-1] := A[j-1] + A[j] od;
for i from 0 to m do T := T, A[i] od;
if display then print(seq(T[i], i=nops([T])-m..nops([T]))) fi;
od; subsop(1=NULL, [T]) end:
swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
# Computes n rows of the triangle:
A163840 := n -> SumTria(swing, n, true);
MATHEMATICA
sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n - k, n - i]*sf[i], {i, k, n}]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)
Triangle interpolating the swinging factorial ( A056040) restricted to even indices with its binomial transform. Same as interpolating bilateral Schroeder paths ( A026375) with the central binomial coefficients ( A000984).
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4
1, 3, 2, 11, 8, 6, 45, 34, 26, 20, 195, 150, 116, 90, 70, 873, 678, 528, 412, 322, 252, 3989, 3116, 2438, 1910, 1498, 1176, 924, 18483, 14494, 11378, 8940, 7030, 5532, 4356, 3432, 86515, 68032, 53538
COMMENTS
For n >= 0, k >= 0 let T(n,k) = sum{i=k..n} binomial(n-k,n-i)*(2i)$ where i$ denotes the swinging factorial of i ( A056040). Triangle read by rows.
EXAMPLE
Triangle begins
1;
3, 2;
11, 8, 6;
45, 34, 26, 20;
195, 150, 116, 90, 70;
873, 678, 528, 412, 322, 252;
3989, 3116, 2438, 1910, 1498, 1176, 924;
MAPLE
Computes n rows of the triangle. For the functions 'SumTria' and 'swing' see A163840. a := n -> SumTria(k->swing(2*k), n, true);
MATHEMATICA
sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)
Binomial transform of the beta numbers 1/beta(n+1,n+1) ( A002457).
+10
2
1, 7, 43, 249, 1395, 7653, 41381, 221399, 1175027, 6196725, 32512401, 169863147, 884318973, 4589954619, 23761814955, 122735222505, 632698778835, 3255832730565, 16728131746145, 85826852897675, 439793834236745, 2251006269442815, 11509340056410735, 58790764269668805
COMMENTS
Also a(n) = Sum_{i=0..n} binomial(n,n-i) (2*i+1)$ where i$ denotes the swinging factorial of i ( A056040).
FORMULA
G.f.: -sqrt(x-1)/(5*x-1)^(3/2).
Recurrence: n*a(n) = (6*n+1)*a(n-1) - 5*(n-1)*a(n-2).
a(n) ~ 4*5^(n-1/2)*sqrt(n)/sqrt(Pi).
(End)
a(n) = hypergeom([3/2, -n], [1], -4) = hypergeom([3/2, n+1], [1], 4/5)/(5*sqrt(5)). - Vladimir Reshetnikov, Apr 25 2016 E.g.f.: exp(3*x) * ((1 + 4*x) * BesselI(0,2*x) + 4 * x * BesselI(1,2*x)). - Ilya Gutkovskiy, Nov 19 2021
MAPLE
a := proc(n) local i; add(binomial(n, i)/Beta(i+1, i+1), i=0..n) end:
MATHEMATICA
CoefficientList[Series[-Sqrt[x-1]/(5*x-1)^(3/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 21 2012 *) sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := Sum[ Binomial[n, n-i]*sf[2*i+1], {i, 0, n}]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jul 26 2013 *)
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