A114347 -id:A114347 - cp's OEIS frontend

A114790 Cumulative product of quintuple factorial A085157.

1, 1, 2, 6, 24, 120, 720, 10080, 241920, 8709120, 435456000, 28740096000, 4828336128000, 1506440871936000, 759246199455744000, 569434649591808000000, 601322989968949248000000

Offset: 0

Jonathan Vos Post, Feb 18 2006

			a(10) = 1!!!!! * 2!!!!! * 3!!!!! * 4!!!!! * 5!!!!! * 6!!!!! * 7!!!!! * 8!!!!! * 9!!!!! * 10!!!!! = 1 * 2 * 3 * 4 * 5 * 6 * 14 * 24 * 36 * 50 = 435456000 = 2^11 * 3^5 * 5^3 * 7.

G. C. Greubel, Table of n, a(n) for n = 0..84
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A000178, A006882, A007662, A085157, A114347.

b:= function(n)
    if n<1 then return 1;
    else return n*b(n-5);
    fi;
  end;
List([0..20], n-> Product([0..n], j-> b(j)) ); # G. C. Greubel, Aug 21 2019

b:= func< n | n eq 0 select 1 else (n lt 6) select n else n*Self(n-5) >;
[(&*[b(j): j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 21 2019

b:= n-> `if`(n < 1, 1, n*b(n-5)); a:= n-> product(b(j), j = 0..n); seq(a(n), n = 0..20); # G. C. Greubel, Aug 21 2019

b[n_]:= If[n<1, 1, n*b[n-5]]; a[n_]:= Product[b[j], {j,0,n}]; Table[a[n], {n,0,20}] (* G. C. Greubel, Aug 21 2019 *)

b(n)=if(n<1, 1, n*b(n-5));
vector(20, n, n--; prod(j=0,n, b(j)) ) \\ G. C. Greubel, Aug 21 2019

@CachedFunction
def b(n):
    if (n<1): return 1
    else: return n*b(n-5)
[product(b(j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 21 2019

a(n) = Product_{j=0..n} A085157(j).
a(n) = n!!!!! * a(n-1) where a(0) = 1, a(1) = 1 and n >= 2.
a(n) = n*(n-5)!!!!! * a(n-1) where a(0) = 1, a(1) = 1, a(2) = 2.

A114779 Cumulative product of quadruple factorial A007662.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 1440, 30240, 967680, 43545600, 5225472000, 1207084032000, 463520268288000, 271159356948480000, 455547719673446400000, 1578472848668491776000000, 9698137182219213471744000000

Offset: 0

Jonathan Vos Post, Feb 18 2006

			a(10) = 1!!!! * 2!!!! * 3!!!! * 4!!!! * 5!!!! * 6!!!! * 7!!!! * 8!!!! * 9!!!! * 10!!!! = 1 * 2 * 3 * 4 * 5 * 12 * 21 * 32 * 45 * 120 = 5225472000 = 2^13 * 3^6 * 5^3 * 7.

Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A006882, A007662, A000178, A114347.

a(n) = Product_{j=0..n} j!!!!.
a(n) = Product_{j=0..n} A007662(j).
a(n) = n!!!! * a(n-1) where a(0) = 1, a(1) = 1 and n >= 2.
a(n) = n*(n-4)!!!! * a(n-1) where a(0) = 1, a(1) = 1, a(2) = 2.

A114796 Cumulative product of sextuple factorial A085158.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 80640, 2177280, 87091200, 4790016000, 344881152000, 31384184832000, 7030057402368000, 2847173247959040000, 1822190878693785600000, 1703748471578689536000000

Offset: 0

Jonathan Vos Post, Feb 18 2006

			a(10) = 1!6 * 2!6 * 3!6 * 4!6 * 5!6 * 6!6 * 7!6 * 8!6 * 9!6 * 10!6
= 1 * 2 * 3 * 4 * 5 * 6 * 7 * 16 * 27 * 40 = 87091200 = 2^11 * 3^5 * 5^2 * 7.
Note that a(10) + 1 = 87091201 is prime, as is a(9) + 1 = 2177281.

G. C. Greubel, Table of n, a(n) for n = 0..91
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A000178, A006882, A007662, A085150, A085157, A085158, A114347.

b:= function(n)
    if n<1 then return 1;
    else return n*b(n-6);
    fi;
  end;
List([0..20], n-> Product([0..n], j-> b(j)) ); # G. C. Greubel, Aug 22 2019

b:=func< n | n le 6 select n else n*Self(n-6) >;
[1] cat [(&*[b(j): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 22 2019

b:= n-> `if`(n<1, 1, n*b(n-5)); a:= n-> product(b(j), j = 0..n); seq(a(n), n = 0..20); # G. C. Greubel, Aug 22 2019

b[n_]:= b[n]= If[n<1, 1, n*b[n-6]]; a[n_]:= Product[b[j], {j,0,n}];
Table[a[n], {n, 0, 20}] (* G. C. Greubel, Aug 22 2019 *)

b(n)=if(n<1, 1, n*b(n-6));
vector(20, n, n--; prod(j=0,n, b(j)) ) \\ G. C. Greubel, Aug 22 2019

def b(n):
    if (n<1): return 1
    else: return n*b(n-6)
[product(b(j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 22 2019

a(n) = Product_{j=0..n} j!!!!!!.
a(n) = Product_{j=0..n} j!6.
a(n) = Product_{j=0..n} A085158(j).
a(n) = n!!!!!! * a(n-1) where a(0) = 1, a(1) = 1 and n >= 2.
a(n) = n*(n-6)!!!!!! * a(n-1) where a(0) = 1, a(1) = 1, a(2) = 2.

A114805 Cumulative sum of quintuple factorial numbers n!!!!! (A085157).

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 22, 36, 60, 96, 146, 212, 380, 692, 1196, 1946, 3002, 5858, 11474, 21050, 36050, 58226, 121058, 250226, 480050, 855050, 1431626, 3128090, 6744794, 13409690, 24659690, 42533546, 96820394, 216171626, 442778090, 836528090

Offset: 0

Jonathan Vos Post, Feb 18 2006

a(1) = 2 is prime; a(3) = 7 is prime; a(4) = 11 is prime; and there are no more primes in the sequence. Semiprime values are: a(2) = 4 = 2^2, a(6) = 22, a(10) = 146 = 2 * 73, a(18) = 11474 = 2 * 5737, a(23) = 250226 = 2 * 125113.

			a(10) = 0!5 + 1!5 + 2!5 + 3!5 + 4!5 + 5!5 + 6!5 + 7!5 + 8!5 + 9!5 + 10!5 =
1 + 1 + 2 + 3 + 4 + 5 + 6 + 14 + 24 + 36 + 50 = 146 = 2 * 73.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A007662, A085157, A114347.

b:= function(n)
    if n<1 then return 1;
    else return n*b(n-5);
    fi;
  end;
List([0..40], n-> Sum([0..n], j-> b(j)) ); # G. C. Greubel, Aug 21 2019

b:= func< n | n eq 0 select 1 else (n lt 6) select n else n*Self(n-5) >;
[(&+[b(j): j in [0..n]]): n in [0..40]]; // G. C. Greubel, Aug 21 2019

b:= n-> `if`(n < 1, 1, n*b(n-5)); a:= n-> sum(b(j), j = 0..n); seq(a(n), n = 0..40); # G. C. Greubel, Aug 21 2019

f5[0]=1; f5[n_]:= f5[n]= If[n<=6, n, n f5[n-5]]; Accumulate[f5/@Range[0, 35]] (* Giovanni Resta, Jun 15 2016 *)

b(n)=if(n<1, 1, n*b(n-5));
vector(40, n, n--; sum(j=0,n, b(j)) ) \\ G. C. Greubel, Aug 21 2019

@CachedFunction
def b(n):
    if (n<1): return 1
    else: return n*b(n-5)
[sum(b(j) for j in (0..n)) for n in (0..40)] # G. C. Greubel, Aug 21 2019

a(n) = Sum_{j=0..n} j!5.
a(n) = Sum_{j=0..n} j!!!!!.
a(n) = Sum_{j=0..n} A085157(j).

A108895 Partial sums of quadruple factorial numbers n!!!! (A007662).

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 28, 49, 81, 126, 246, 477, 861, 1446, 3126, 6591, 12735, 22680, 52920, 118755, 241635, 450480, 1115760, 2629965, 5579085, 10800210, 28097490, 68981025, 151556385, 302969010, 821887410, 2089276995, 4731688515

Offset: 0

Jonathan Vos Post, Feb 08 2006

Quadruple factorial numbers n!!!! = n*(n-4)!!!!, 0!!!! = 1!!!! = 1, 2!!!! = 2, 3!!!! = 3. The cumulative sum a(n) is prime for n = 1, 3, 4 and never again, as all values from a(8) = 81 are multiples of 3. The cumulative sum a(n) is semiprime for n = 2, 7 and never again, as all values from a(16) are divisible by both 3 and 5.

			a(31) = 1 + 1 + 2 + 3 + 4 + 5 + 12 + 21 + 32 + 45 + 120 + 231 + 384 + 585 + 1680 + 3465 + 6144 + 9945 + 30240 + 65835 + 122880 + 208845 + 665280 + 1514205 + 2949120 + 5221125 + 17297280 + 40883535 + 82575360 + 151412625 + 518918400 + 1267389585 = 2089276995 = 3 * 5 * 13 * 337 * 31793.

J. Spanier and K. B. Oldham, An Atlas of Functions, Hemisphere, NY, 1987, p. 23.

Cf. A000040, A001358, A007662, A114347.

NFactorialM[n_Integer, m_Integer] := Block[{k = n, p = Max[1, n]}, While[k > m, k -= m; p *= k]; p]; Table[ Sum[ NFactorialM[i, 4], {i, 0, n}], {n, 0, 33}] (* Robert G. Wilson v, Feb 21 2006 *)

a(n) = Sum_{i=0..n} i!!!!.

a(n) = Sum_{i=0..n} A007662(i).

A114778 Cumulative product of triple factorial A007661.

Original entry on oeis.org

1, 1, 2, 6, 24, 240, 4320, 120960, 9676800, 1567641600, 438939648000, 386266890240000, 750902834626560000, 2733286318040678400000, 33674087438261157888000000, 981936389699695364014080000000

Offset: 0

Jonathan Vos Post, Feb 18 2006

This is the product analog of what for sums is A114347.

			a(10) = 1!!! * 2!!! * 3!!! * 4!!! * 5!!! * 6!!! * 7!!! * 8!!! * 9!!! * 10!!!
= 1 * 2 * 3 * 4 * 10 * 18 * 28 * 80 * 162 * 280 = 438939648000 = 2^15 * 3^7 * 5^3 * 7^2.

Eric Weisstein's World of Mathematics, Multifactorial.

Cf. A006882, A000178, A114347.

a(n) = Product_{j=0..n} j!!!. a(n) = Product_{j=0..n} A007661(j). a(n) = n!!! * a(n-1) where a(0) = 1, a(1) = 1 and n >= 2. a(n) = n*(n-3)!!! * a(n-1) where a(0) = 1, a(1) = 1, a(2) = 2 and n >= 3.

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A114790 Cumulative product of quintuple factorial A085157.

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A114779 Cumulative product of quadruple factorial A007662.

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A114796 Cumulative product of sextuple factorial A085158.

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A114805 Cumulative sum of quintuple factorial numbers n!!!!! (A085157).

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A108895 Partial sums of quadruple factorial numbers n!!!! (A007662).

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A114778 Cumulative product of triple factorial A007661.

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