A006882 -id:A006882 - cp's OEIS frontend

A263529 Binomial transform of double factorial n!! (A006882).

1, 2, 5, 13, 37, 111, 355, 1191, 4201, 15445, 59171, 234983, 966397, 4101709, 17946783, 80754331, 373286481, 1769440513, 8592681907, 42689422871, 216789872741, 1124107246669, 5947013363479, 32071798826115, 176194545585529, 985330955637801, 5605802379087067

Offset: 0

Vladimir Reshetnikov, Oct 19 2015

nonn

			G.f. = 1 + 2*x + 5*x^2 + 13*x^3 + 37*x^4 + 111*x^5 + 355*x^6 + 1191*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..795
Eric Weisstein's MathWorld, Double Factorial.

Cf. A006882, A262020.

Table[Sum[k!!*Binomial[n, k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 20 2015 *)

vector(50, n, n--; sum(k=0, n, prod(i=0, (k-1)\2, k - 2*i)*binomial(n,k))) \\ Altug Alkan, Oct 20 2015

a(n) = Sum_{k=0..n} k!!*binomial(n,k), where k!! = A006882(k).
Sum_{k=0..n} (-1)^(k+n)*a(k)*binomial(n,k) = n!!.
E.g.f.: exp(x) + exp((2*x+x^2)/2)*(2 + sqrt(2*Pi)*erf(x/sqrt(2)))*x/2.
Recurrence: (n+1)*a(n+2) = (n+2)*a(n+1) + (n+1)*(n+2)*a(n) - 1.
a(n) ~ (sqrt(2) + sqrt(Pi))/2 * n^(n/2 + 1/2) * exp(sqrt(n) - n/2 - 1/4). - Vaclav Kotesovec, Oct 20 2015
0 = a(n)*(+a(n+1) - 2*a(n+2) - 2*a(n+3) + a(n+4)) + a(n+1)*(+3*a(n+2) + a(n+3) - a(n+4)) + a(n+2)*(-2*a(n+2) + a(n+3)) for all n>=0. - Michael Somos, Oct 20 2015
G.f.: Sum_{k>=0} k!!*x^k/(1 - x)^(k+1). - Ilya Gutkovskiy, Apr 12 2019

A095175 Denominator of b(n) given by b(1) = 1, b(2) = 2; for n >= 3, b(n) = (-1)^n (2n-1) ((n-2)!!)^2/((n-1)!!)^2, where n!! is the double factorial A006882.

Original entry on oeis.org

1, 1, 4, 9, 64, 225, 256, 245, 16384, 99225, 65536, 480249, 1048576, 1002001, 4194304, 41409225, 1073741824, 2393453205, 4294967296, 4102737925, 68719476736, 940839860961, 274877906944, 4113258565689, 17592186044416, 16802526820625, 70368744177664

Offset: 1

N. J. A. Sloane, based on a suggestion of Leroy Quet, Jul 03 2004

A095159(2n)/a(2n) -> pi as n -> inf.; A095159(2n+1)/a(2n+1) -> -4/pi as n -> inf. - Leroy Quet, Aug 03 2004

Cf. A006882, A095159.

A114338 Number of divisors of n!! (double factorial = A006882(n)).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 10, 8, 16, 16, 36, 32, 66, 64, 144, 120, 192, 240, 340, 480, 570, 864, 1200, 1728, 1656, 2880, 3456, 4320, 5616, 8640, 9072, 17280, 10752, 28800, 22176, 46080, 30240, 92160, 62208, 152064, 84240, 304128, 128000, 608256, 201600

Offset: 0

Giovanni Resta, Feb 07 2006

It appears that a(n+2) = 2*a(n) if n is in A238526. - Michel Lagneau, Dec 07 2015

			a(5) = 4 since 5!! = 15 and the divisors are 1, 3, 5 and 15.
a(6) = 10 because 6!! = A006882(6) = 48 has precisely ten distinct divisors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. - _Michel Lagneau_, Dec 07 2016

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000005, A006882, A027423, A238526.

f := proc(n)
numtheory[tau](doublefactorial(n)) ;
end proc: # R. J. Mathar, Dec 14 2015

Mathematica
```
DivisorSigma[0,Range[50]!! ]
```

df(n) = if( n<0, 0, my(E); E = exp(x^2 / 2 + x * O(x^n)); n! * polcoeff( 1 + E * x * (1 + intformal(1 / E)), n)); \\ A006882
vector(100, n, n--; numdiv(df(n))) \\ Altug Alkan, Dec 07 2015

a(n) = sigma_0(n!!) = tau(n!!) = A000005(A006882(n)).

A115648 Square numbers which are the sum of distinct double factorials (A006882).

Original entry on oeis.org

1, 4, 9, 16, 25, 49, 64, 121, 169, 400, 441, 961, 1444, 3844, 3969, 4225, 4356, 4900, 5184, 10404, 11449, 11881, 14400, 15625, 47089, 47524, 56644, 57600, 139129, 145924, 149769, 182329, 192721, 695556, 705600, 792100, 837225, 2073600

Offset: 1

Giovanni Resta, Jan 28 2006

nonn

Double factorials 0!! and 1!! are not considered distinct. Note that double factorial (n!!) is different from (n!)!.

			10404 = 102^2 = 1!! + 4!! + 11!!.

Cf. A006882, A115649, A115650, A115651, A025494.

A115649 Numbers whose square is the sum of distinct double factorials (A006882).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 13, 20, 21, 31, 38, 62, 63, 65, 66, 70, 72, 102, 107, 109, 120, 125, 217, 218, 238, 240, 373, 382, 387, 427, 439, 834, 840, 890, 915, 1440, 1474, 1638, 1650, 3213, 3220, 3241, 3332, 3540, 5886, 6751, 6855, 13730, 14078, 14911

Offset: 1

Giovanni Resta, Jan 28 2006

nonn

Double factorials 0!! and 1!! are not considered distinct. Note that double factorial (n!!) is different from (n!)!.

			102^2 = 1!! + 4!! + 11!!.

Cf. A006882, A115648, A115650.

A115650 Triangular numbers which are the sum of distinct double factorials (A006882).

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 66, 105, 120, 153, 171, 435, 561, 946, 1378, 1485, 3916, 4005, 4278, 4851, 4950, 11781, 14365, 15576, 47586, 50086, 51360, 57970, 60378, 60726, 61425, 61776, 139128, 145530, 146070, 149878, 150426, 182710, 659526, 702705

Offset: 1

Giovanni Resta, Jan 28 2006

nonn

Double factorials 0!! and 1!! are not considered distinct. Note that double factorial (n!!) is different from (n!)!.

			T(527) = 19128 = 13!! + 10!! + 7!! + 6!!.

Cf. A006882, A115648, A115649.

A202212 Triangle read by rows: T(n,k) (1 <= k <= n-1, n >= 2) = d(2(n-k)-1)(d(2n-2)/d(2(n-k)-2) - d(2n-3)/d(2(n-k)-3)), where d = A006882 is the double factorial function.

Original entry on oeis.org

1, 3, 5, 15, 27, 33, 105, 195, 261, 279, 945, 1785, 2475, 2925, 2895, 10395, 19845, 28035, 34425, 37935, 35685, 135135, 259875, 371385, 465255, 533925, 562275, 509985, 2027025, 3918915, 5644485, 7158375, 8390025, 9218475, 9401805, 8294895, 34459425, 66891825, 96891795, 123898005, 147093975, 165209625, 176067675, 175313565, 151335135, 654729075, 1274998725, 1854727875, 2385808425, 2857013775, 3252014325, 3545408475, 3693650625, 3609649575, 3061162125

Offset: 2

N. J. A. Sloane, Dec 14 2011

			Triangle begins
1,
3, 5,
15, 27, 33,
105, 195, 261, 279,
945, 1785, 2475, 2925, 2895,
10395, 19845, 28035, 34425, 37935, 35685,
135135, 259875, 371385, 465255, 533925, 562275, 509985,
...

N. Ochiumi, On the total sum of number of nodes covering a given number of leaves in an unordered binary tree.

Edges of triangle are A006882 and A129890.

d:=doublefactorial;
a:=(n,k)-> d(2*(n-k)-1)*(d(2*n-2)/d(2*(n-k)-2) - d(2*n-3)/d(2*(n-k)-3));
f:=n->[seq(a(n,k),k=1..n-1)];
for n from 1 to 10 do lprint(f(n)); od:

a[n_, k_] := (2*(n-k)-1)!!*((2*n-2)!!/(2*(n-k)-2)!!-(2*n-3)!!/(2*(n-k)-3)!!); Table[a[n, k], {n, 2, 11}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)

A232788 A232773(n) / A006882(n): Permanent of the n X n matrix with elements [1,2,...,n^2], divided by n!!.

Original entry on oeis.org

1, 1, 5, 150, 6932, 965380, 143299890, 51176650000, 16737737386944, 11806879466638656, 7023172771916784000, 8447153882019234307200, 8134080139379917205277696, 15176253254155788712392633600, 21875035292051870323313614135440, 59270306784445546617788929301760000

Offset: 0

M. F. Hasler, Nov 30 2013

nonn

Limit n->infinity a(n)^(1/n)/n^(5/2) = exp(-3/2). - Vaclav Kotesovec, Nov 08 2014

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A232773, A006882.

with(combinat):
a:= n-> (-1)^n *add(n^k *stirling1(n, n-k)*stirling1(n+1, k+1)
        *(n-k)!* k!, k=0..n)/doublefactorial(n):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 02 2013

Flatten[{1,Table[(-1)^n*Sum[n^k*StirlingS1[n,n-k]*StirlingS1[n+1,k+1]*(n-k)!*k!,{k,0,n}]/n!!,{n,1,20}]}] (* Vaclav Kotesovec, Nov 08 2014 *)

n->(-1)^n*sum(k=0,n,n^k*stirling(n,n-k)*stirling(n+1,k+1)*(n-k)!*k!)/A006882(n)

a(0)=1 inserted by Alois P. Heinz, Dec 02 2013

A262020 Inverse binomial transform of double factorial n!! = A006882(n).

Original entry on oeis.org

1, 0, 1, -1, 5, -11, 43, -127, 489, -1693, 6771, -26071, 109693, -457757, 2028671, -9039931, 42101329, -198411489, 967906675, -4791497559, 24401815141, -126243354637, 669094876055, -3603105436163, 19818039219577, -110721426757801, 630419303537115

Offset: 0

Alois P. Heinz, Oct 22 2015

sign

Alois P. Heinz, Table of n, a(n) for n = 0..800

Cf. A000166 (the same for n!), A006882, A263529.

a:= proc(n) option remember; `if`(n<3, (n-1)^2,
       (n-2)*a(n-3) +(n-1)*a(n-2) -2*a(n-1))
    end:
seq(a(n), n=0..30);

Table[Sum[(-1)^(n-k) * Binomial[n, k] * k!!, {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 31 2017 *)

E.g.f.: (2*exp(x^2/2)*x+2+sqrt(2*Pi)*exp(x^2/2)*erf(x/sqrt(2))*x) / (2*exp(x)).
a(n) = (n-2)*a(n-3)+(n-1)*a(n-2)-2*a(n-1) for n>2, a(n) = (n-1)^2 otherwise.
a(n) = Sum_{k=0..n} (-1)^k * C(n,k) *  A006882(n-k).
a(n) ~ (-1)^n * (sqrt(Pi) - sqrt(2)) * exp(sqrt(n) - n/2 - 1/4) * n^((n+1)/2) / 2. - Vaclav Kotesovec, Oct 31 2017
G.f.: Sum_{k>=0} k!!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 12 2019

A306184 a(n) = (2n+1)!! mod (2n)!! where k!! = A006882(k).

Original entry on oeis.org

1, 7, 9, 177, 2715, 42975, 91665, 3493665, 97345395, 2601636975, 70985324025, 57891366225, 9411029102475, 476966861546175, 20499289200014625, 847876038362978625, 35160445175104123875, 1487419121780448231375, 945654757149212735625, 357657177058846280240625

Offset: 1

Alex Ratushnyak, Jan 27 2019

nonn

a(n) is divisible by A049606(n). - Robert Israel, Jan 28 2019

			a(3) = A006882(7) mod A006882(6) = (7*5*3) mod (6*4*2) = 105 mod 48 = 9.

Robert Israel, Table of n, a(n) for n = 1..403

Cf. A006882, A049606, A122649, A129890, A232617, A232618, A306185.

f:= n -> doublefactorial(2*n+1) mod doublefactorial(2*n):
map(f, [$1..40]); # Robert Israel, Jan 28 2019

Mod[#[[2]],#[[1]]]&/@Partition[Range[2,42]!!,2] (* Harvey P. Dale, May 29 2025 *)

o=e=1
for n in range(2, 99, 2):
  o*=n+1
  e*=n
  print(o%e, end=', ')

a(n) = A006882(2*n+1) mod A006882(2*n).

cp's OEIS Frontend

A263529 Binomial transform of double factorial n!! (A006882).

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A095175 Denominator of b(n) given by b(1) = 1, b(2) = 2; for n >= 3, b(n) = (-1)^n (2n-1) ((n-2)!!)^2/((n-1)!!)^2, where n!! is the double factorial A006882.

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A114338 Number of divisors of n!! (double factorial = A006882(n)).

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A115648 Square numbers which are the sum of distinct double factorials (A006882).

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A115649 Numbers whose square is the sum of distinct double factorials (A006882).

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A115650 Triangular numbers which are the sum of distinct double factorials (A006882).

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A202212 Triangle read by rows: T(n,k) (1 <= k <= n-1, n >= 2) = d(2*(n-k)-1)*(d(2*n-2)/d(2*(n-k)-2) - d(2*n-3)/d(2*(n-k)-3)), where d = A006882 is the double factorial function.

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A232788 A232773(n) / A006882(n): Permanent of the n X n matrix with elements [1,2,...,n^2], divided by n!!.

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Maple

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Extensions

A262020 Inverse binomial transform of double factorial n!! = A006882(n).

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A202212 Triangle read by rows: T(n,k) (1 <= k <= n-1, n >= 2) = d(2(n-k)-1)(d(2n-2)/d(2(n-k)-2) - d(2n-3)/d(2(n-k)-3)), where d = A006882 is the double factorial function.