A007681
a(n) = (2*n+1)^2*n!.
Original entry on oeis.org
1, 9, 50, 294, 1944, 14520, 121680, 1134000, 11652480, 130999680, 1600300800, 21115987200, 299376000000, 4539498163200, 73316942899200, 1256675067648000, 22784918188032000, 435717099417600000
Offset: 0
- H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A007682
a(n) = -Sum_{k = 0..n-1} (n+k)!a(k)/(2k)!.
Original entry on oeis.org
1, -1, 1, 1, -1, -17, -107, -415, 1231, 56671, 924365, 11322001, 97495687, -78466897, -31987213451, -1073614991039, -26754505127713, -558657850929473, -9259584394031075, -70982644052430799, 3334438016903221111, 240585292388924690959, 10679411902033402697861
Offset: 0
- H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
-
A007682 := proc(n) option remember; if n=0 then RETURN(1) fi; if n>0 then RETURN((-1)*add((n+k)!*'A007682(k)'/(2*k)!, k=0..n-1 )) fi; end;
-
a[n_] := a[n] = -Sum[(n+k)!*a[k]/(2*k)!, {k, 0, n-1}]; a[0] = 1; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Jan 27 2014 *)
A082028
Expansion of exp(x)*(1+x)/(1-x)^2.
Original entry on oeis.org
1, 4, 17, 82, 457, 2936, 21529, 178102, 1644017, 16768972, 187417921, 2278607354, 29947410937, 423169937152, 6398329449737, 103084196690206, 1763095226149729, 31906336189354772, 609120963614954737
Offset: 0
-
CoefficientList[Series[E^x*(1+x)/(1-x)^2, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jul 02 2015 *)
A084253
a(n) is the denominator of the coefficient of z^(2n-1) in the Maclaurin expansion of Sqrt[Pi]Erfi[z].
Original entry on oeis.org
1, 3, 5, 21, 108, 660, 4680, 37800, 342720, 3447360, 38102400, 459043200, 5987520000, 84064780800, 1264085222400, 20268952704000, 345226033152000, 6224529991680000, 118443913555968000, 2372079457972224000
Offset: 1
-
Join[{1, 3}, Table[(2*n - 1)*n!/(2*n), {n,3,50}]] (* or *) Denominator[ CoefficientList[Series[Sqrt[Pi]*Erf[t], {t, 0, 10}], t]][[2 ;; ;; 2]] (* G. C. Greubel, Jan 12 2017 *)
-
concat([1,3], for(n=3, 50, print1((2*n-1)*n!/(2*n), ", "))) \\ G. C. Greubel, Jan 12 2017
A154987
Triangle read by rows: T(n,k) = t(n,k) + t(n,n-k), where t(n,k) = 2*(n!/k!)*(2*(n + k) - 1).
Original entry on oeis.org
-2, 4, 4, 13, 20, 13, 41, 69, 69, 41, 183, 268, 264, 268, 183, 1099, 1405, 1080, 1080, 1405, 1099, 7943, 9486, 5970, 4080, 5970, 9486, 7943, 65547, 75775, 43806, 20370, 20370, 43806, 75775, 65547, 604831, 685672, 384552, 149520, 77280, 149520, 384552, 685672, 604831
Offset: 0
-2;
4, 4;
13, 20, 13;
41, 69, 69, 41;
183, 268, 264, 268, 183;
1099, 1405, 1080, 1080, 1405, 1099;
7943, 9486, 5970, 4080, 5970, 9486, 7943;
65547, 75775, 43806, 20370, 20370, 43806, 75775, 65547;
...
-
t:= proc(n,k) option remember; ## simplified t;
2*(n+k-1/2)*(n!/k!);
end proc:
A154987:= proc(n,k) ## n >= 0 and k = 0 .. n
t(n,k) + t(n,n-k)
end proc: # Yu-Sheng Chang, Apr 13 2020
-
(* First program *)
t[n_, k_]:= 2*n!*Gamma[n+k+1/2]/(k!*Gamma[n+k-1/2]);
T[n_, k_]:= t[n, k] + t[n,n-k];
Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten
(* Second Program *)
T[n_, k_]:= Binomial[n, k]*((n-k)!*(2*n+2*k-1) + k!*(4*n-2*k-1));
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, May 28 2020 *)
-
def T(n, k): return binomial(n, k)*(factorial(n-k)*(2*n+2*k-1) + factorial(k)*(4*n-2*k-1))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 28 2020
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