A001142 -id:A001142 - cp's OEIS frontend

A296591 a(n) = Product_{k=0..n} (n + k)!.

1, 2, 288, 12441600, 421382062080000, 23120161750363668480000000, 3683853104727992382799761899520000000000, 2777528195026874073410445622205453260145295360000000000000

Offset: 0

Vaclav Kotesovec, Dec 16 2017

nonn

Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function

Cf. A001142, A007685, A086205, A098694, A110131, A112332, A268196, A296589, A296590.

a:= proc(n) option remember; `if`(n=0, 1,
      a(n-1) *(2*n-1)! *(2*n)! /(n-1)!)
    end:
seq(a(n), n=0..7);  # Alois P. Heinz, Jul 11 2024

Table[Product[(n + k)!, {k, 0, n}], {n, 0, 10}]
Table[Product[(2*n - k)!, {k, 0, n}], {n, 0, 10}]
Table[BarnesG[2*n + 2]/BarnesG[n + 1], {n, 0, 10}]

a(n) = BarnesG(2*n + 2) / BarnesG(n + 1).

a(n) ~ 2^(2*n^2 + 5*n/2 + 11/12) * n^((n+1)*(3*n+1)/2) * Pi^((n+1)/2) / exp(9*n^2/4 + 2*n).

Missing a(0)=1 inserted by Georg Fischer, Nov 18 2021

A249433 Integers n such that n! does not divide the product of elements on row n of Pascal's triangle.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 37, 38, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 80, 81, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 97, 98, 99, 101, 103, 105, 109, 110, 111, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 127, 128

Offset: 1

Antti Karttunen, Nov 02 2014

nonn

Integers n such that A249151(n) < n.

Equally: Integers n such that A249431(n) is negative.

			See the examples at A249434.

Antti Karttunen, Table of n, a(n) for n = 1..2676

Complement: A249434.
Subsequences: A000225, A024023, A024049, etc., (after their two initial terms, i.e. A249435 without its initial zero is also a subsequence), A249424, A249436.
Cf. A000142, A001142, A007318, A249151, A249431.

A056606 Squarefree kernel of lcm(binomial(n,0), ..., binomial(n,n)).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 30, 105, 70, 42, 210, 2310, 2310, 4290, 6006, 15015, 30030, 170170, 510510, 1939938, 1385670, 881790, 9699690, 223092870, 44618574, 17160990, 74364290, 31870410, 223092870, 6469693230, 6469693230, 100280245065

Offset: 0

Labos Elemer, Aug 07 2000

nonn

Also squarefree kernel of A001142; row products in table A256113. - Reinhard Zumkeller, Mar 21 2015
a(2372) has 1001 decimal digits. - Michael De Vlieger, Jul 14 2017
Also the squarefree kernel of the cumulative product of n^n/n!. - Peter Luschny, Dec 21 2019
Conjecture: the few odd values belong to A070826. - Bill McEachen, Jun 24 2023
And their indices appear to be A007053. - Michel Marcus, Jul 01 2023

			a(7) = 105 because lcm(1, 7, 21, 35) = 105 is already squarefree.
a(0) = 1 because n^n/n! = 1 for the integer n = 0. - _Peter Luschny_, Dec 21 2019

Michael De Vlieger, Table of n, a(n) for n = 0..2371

Cf. A002944, A034386, A048633, A003418, A000142, A001405.
Cf. A007947, A001142, A256113, A002109, A000178, A329947.
Cf. A070826.

a056606 = a007947 . a001142  -- Reinhard Zumkeller, Mar 21 2015

h := n -> mul(k^k/factorial(k), k=0..n):
rad := n -> mul(k, k = numtheory[factorset](n)):
seq(rad(h(n)), n=0..31); # Peter Luschny, Dec 21 2019

Table[Apply[Times, FactorInteger[Product[k^(2 k - 1 - n), {k, n}]][[All, 1]]], {n, 0, 31}] (* or *)
Table[Apply[Times, FactorInteger[Apply[LCM, Range@ n]/n][[All, 1]]], {n, 1, 32}] (* Michael De Vlieger, Jul 14 2017 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = rad(lcm(vector(n+1, k, binomial(n,k-1)))); \\ Michel Marcus, Jun 24 2023

a(n) = A007947(A002944(n+1)). - Michel Marcus, Dec 21 2019

a(n) = radical(hyperfactorial(n)/superfactorial(n)) = A007947(A002109(n)/ A000178(n)) for n >= 0. - Peter Luschny, Dec 21 2019

Extended with a(0) = 1 by Peter Luschny, Dec 21 2019

A004788 Number of distinct prime divisors of the numbers in row n of Pascal's triangle.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 9, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 13, 13, 14, 13, 14, 15, 14, 14, 14, 14, 15, 15, 15, 16, 15, 15, 16, 17, 17, 17, 18, 17, 17, 17, 18, 18, 18, 19, 19, 20, 20

Offset: 0

Clark Kimberling

nonn

Also the number of prime divisors of A002944(n) = lcm_{j=0..floor(n/2)} binomial(n,j).
The terms are increasing by intervals, then decrease once. The local maxima are obtained for 23, 44, 47, 55, 62, 79, 83, 89, 104, 119, 131, 134, 139, 143, .... - Michel Marcus, Mar 21 2013
a(A004789(n)) = n and a(m) != n for m < A004789(n). - Reinhard Zumkeller, Mar 16 2015

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A004789.

Cf. A001221, A001142, A256113.

a004788 = a001221 . a001142  -- Reinhard Zumkeller, Mar 16 2015

Table[prd = Product[Binomial[n, k], {k, 0, n}]; If[prd == 1, 0, Length[FactorInteger[prd]]], {n, 0, 100}] (* T. D. Noe, Mar 21 2013 *)

a(n) = {sfp = Set(); for (k=1, n-1, sfp = setunion(sfp, Set(factor(binomial(n, k))[,1]))); return (length(sfp));} \\ Michel Marcus, Mar 21 2013

a(n) = A001221(A001142(n)). - Reinhard Zumkeller, Mar 16 2015

A051459 Number of orderings of the subsets of a set with n elements that are compatible with the subsets' sizes; i.e., if A, B are two subsets with A <= B then Card(A) <= Card(B).

Original entry on oeis.org

1, 1, 2, 36, 414720, 189621927936000000, 2156695499113014719143826715127578624000000000000

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 15 2003

nonn

a(7) has 127 digits and too large to include in sequence. - Ray Chandler, Nov 22 2003
From Valentin Bakoev, Nov 20 2017, May 17 2019: (Start)
a(n) is the number of possible orderings of the vectors of the n-dimensional Boolean cube (hypercube) {0,1}^n in accordance with their (Hamming) weights. For arbitrary vectors u, v of {0, 1}^n, if wt(u)a(n) is also the number of all possible topological orders (sortings) of the directed acyclic graph (DAG) defined by the same poset: {0,1}^n and the relation weight order as it is defined and explained above.
Both comments correspond to the name of the sequence since the corresponding Boolean algebras are isomorphic. (End)

Michael De Vlieger, Table of n, a(n) for n = 0..9
Valentin Bakoev, Some problems and algorithms related to the weight order relation on the n-dimensional Boolean cube, Discrete Mathematics, Algorithms and Applications, Vol. 13 No 3, 2150021 (2021); arXiv preprint, arXiv:1811.04421 [cs.DM], 2018-2020.

Cf. A000722, A001142, A022914, A294648.

a:= n-> mul(binomial(n, i)!, i=0..n):
seq(a(n), n=0..6);  # Alois P. Heinz, Nov 20 2017

Array[Product[Binomial[#, i]!, {i, #}] &, 7, 0] (* Michael De Vlieger, Nov 20 2017 *)

a(n):= prod(binomial(n,k)!,k,0,n); /* Valentin Bakoev, May 17 2019 */

a(n) = prod(k=0, n, binomial(n, k)!); \\ Michel Marcus, May 18 2019

a(n) = C(n, 0)! * C(n, 1)! * C(n, 2)! * ... * C(n, n)! = A000722(n) / A022914(n).

log(a(n)) ~ log(2) * n * 2^n. - Vaclav Kotesovec, Nov 24 2023

More terms from Ray Chandler, Nov 22 2003

a(0)=1 prepended by Alois P. Heinz, Nov 20 2017

A249345 The exponent of the highest power of 5 dividing the product of the elements on the n-th row of Pascal's triangle.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 3, 2, 1, 0, 8, 6, 4, 2, 0, 12, 9, 6, 3, 0, 16, 12, 8, 4, 0, 44, 38, 32, 26, 20, 43, 36, 29, 22, 15, 42, 34, 26, 18, 10, 41, 32, 23, 14, 5, 40, 30, 20, 10, 0, 88, 76, 64, 52, 40, 82, 69, 56, 43, 30, 76, 62, 48, 34, 20, 70, 55, 40, 25, 10, 64, 48, 32, 16, 0

Offset: 0

Antti Karttunen, Oct 28 2014

nonn

Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..3124 from Antti Karttunen).
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.

Row 3 of array A249421.

Cf. A001142, A007318, A112765, A187059, A249343, A249347.

A249345[n_] := Sum[#*((#+1)*5^k - n - 1) & [Floor[n/5^k]], {k, Floor[Log[5, n]]}];
Array[A249345, 100, 0] (* Paolo Xausa, Feb 11 2025 *)

allocatemem(234567890);
A249345(n) = sum(k=0, n, valuation(binomial(n, k), 5));
for(n=0, 3124, write("b249345.txt", n, " ", A249345(n)));

(define (A249345 n) (A112765 (A001142 n)))

(define (A249345 n) (add (lambda (n) (A112765 (A007318 n))) (A000217 n) (A000096 n)))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (+ 1 i) (+ res (intfun i)))))))

a(n) = A112765(A001142(n)).
a(n) = Sum_{k=0..n} A112765(binomial(n,k)).
a(n) = Sum_{i=1..n} (2*i-n-1)*v_5(i), where v_5(i) = A112765(i) is the exponent of the highest power of 5 dividing i. - Ridouane Oudra, Jun 02 2022
a(n) = Sum_{k=1..floor(log_5(n))} t*((t+1)*5^k - n - 1), where t = floor(n/(5^k)). - Paolo Xausa, Feb 11 2025, derived from Ridouane Oudra's formula above.

A249347 The exponent of the highest power of 7 dividing the product of the elements on the n-th row of Pascal's triangle.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 5, 4, 3, 2, 1, 0, 12, 10, 8, 6, 4, 2, 0, 18, 15, 12, 9, 6, 3, 0, 24, 20, 16, 12, 8, 4, 0, 30, 25, 20, 15, 10, 5, 0, 36, 30, 24, 18, 12, 6, 0, 90, 82, 74, 66, 58, 50, 42, 89, 80, 71, 62, 53, 44, 35, 88, 78, 68, 58, 48, 38, 28, 87, 76, 65, 54, 43, 32, 21, 86, 74, 62, 50, 38, 26, 14, 85, 72, 59, 46, 33, 20, 7, 84, 70, 56, 42, 28, 14, 0

Offset: 0

Antti Karttunen, Oct 28 2014

nonn

Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..2400 from Antti Karttunen).
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.

Row 4 of array A249421.

Cf. A001142, A007318, A214411, A187059, A249343, A249345.

A249347[n_] := Sum[#*((#+1)*7^k - n - 1) & [Floor[n/7^k]], {k, Floor[Log[7, n]]}];
Array[A249347, 100, 0] (* Paolo Xausa, Feb 08 2025 *)

allocatemem(234567890);
A249347(n) = sum(k=0, n, valuation(binomial(n, k), 7));
for(n=0, 2400, write("b249347.txt", n, " ", A249347(n)));

(define (A249347 n) (A214411 (A001142 n)))

(define (A249347 n) (add (lambda (n) (A214411 (A007318 n))) (A000217 n) (A000096 n)))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (+ 1 i) (+ res (intfun i)))))))

a(n) = A214411(A001142(n)).
a(n) = Sum_{k=0..n} A214411(binomial(n,k)).
a(n) = Sum_{i=1..n} (2*i-n-1)*v_7(i), where v_7(i) = A214411(i) is the exponent of the highest power of 7 dividing i. - Ridouane Oudra, Jun 03 2022
a(n) = Sum_{k=1..floor(log_7(n))} t*((t+1)*7^k - n - 1), where t = floor(n/(7^k)). - Paolo Xausa, Feb 09 2025, derived from Ridouane Oudra's formula above.

A073225 a(n) = ceiling(n^n/n!).

Original entry on oeis.org

1, 1, 2, 5, 11, 27, 65, 164, 417, 1068, 2756, 7148, 18614, 48639, 127464, 334865, 881658, 2325751, 6145597, 16263867, 43099805, 114356612, 303761261, 807692035, 2149632062, 5726042116, 15264691108, 40722913455, 108713644517

Offset: 0

Michael Somos, Jul 22 2002

nonn

The van der Waerden conjecture, now a theorem thanks to Egorycev, states that the permanent of any n X n doubly stochastic matrix is >= n!/n^n, with equality iff the matrix has all entries equal to 1/n.
Therefore the reciprocal of the permanent of any n X n doubly stochastic matrix is bounded from above by n^n/n! and this sequence.
n^n/n! = A001142(n)/A001142(n-1), where A001142(n) is product{k=0 to n} C(n,k) (where C() is a binomial coefficient). - Leroy Quet, May 01 2004

			G.f.: 1 + x + 2*x^2 + 5*x^3 + 11*x^4 + 27*x^5 + 65*x^6 + 164*x^7 + 417*x^8 + ...

G. P. Egorycev, Solution of the van der Waerden problem for permanents (Russian), Preprint IFSO-13 M. Akad. Nauk SSSR Sibirsk. Otdel., Inst. Fiz., Krasnoyarsk, 1980. 12 pp. Math. Rev. 82e:15006.
J. H. van Lint, R. M. Wilson, A Course in Combinatorics, Cambridge Univ. Press, 1992. p. 86.

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

Cf. A055775, A094082.

[Ceiling(n^n/Factorial(n)): n in [0..50]]; // G. C. Greubel, May 29 2018

Join[{1}, Table[Ceiling[n^n/n!], {n,1,50}]] (* G. C. Greubel, May 29 2018 *)

PARI
```
{a(n) = ceil(n^n / n!)}
```

A208650 Number of constant paths through the subset array of {1,2,...,n}; see Comments.

Original entry on oeis.org

1, 2, 6, 36, 480, 15000, 1134000, 211768200, 99131719680, 117595223746560, 356467003200000000, 2779532232516963000000, 56049508602150185041920000, 2935889842347365340037522521600

Offset: 1

Clark Kimberling, Mar 01 2012

nonn

Let I(n)={1,2,...,n}. Arrange the subsets of I(n) in an array S(n) of n rows, where row k consists of all the numbers in all the k-element subsets, including repetitions. Each i in I(n) occurs C(n-1,k-1) times in row k of S(n); index these occurrences as
...
(k,1,1),(k,1,2),...,(k,1,r),(k,2,1),...,(k,2,r),...,(k,n,1),...,(k,n,r),
...
where r=C(n-1,k-1).
Definitions:
(1) A path through I(n) is an n-tuple of triples, ((1,i(1),j(1)), (2,i(2),j(2)), ..., (n,i(n),j(n))), formed from the above indexing of the numbers in S(n).
(2) The trace of such a path p is the n-tuple (i(1),i(2),...,i(n)).
(3) The range of p is the set {i(1),i(2),...,i(n)}.
(4) Path p has property P if its trace or range has property P.
...
Guide to sequences which count paths according to selected properties:
property................................sequence
range = {1}.............................A001142(n-1)
constant (range just one element).......A208650
range = {1,2,...,n}.....................A208651
palindromic.............................A208654
palindromic with i(1)=1.................A208655

			Taking n=3:
row 1:  {1},{2},{3} ---------> 1,2,3
row 2:  {1,2},{1,3},{2,3} ---> 1,1,2,2,3,3
row 3:  {1,2,3} -------------> 1,2,3
3 ways to choose a number from row 1,
2 ways to choose same number from row 2,
1 way to choose same number from row 3.
Total:  a(3) = 1*2*3 = 6 paths.

Cf. A208651.

p[n_]:=Product[Binomial[n-1,k],{k,1,n-1}]
Table[p[n],{n,1,20}]    (* A001142(n-1) *)
Table[p[n]*n,{n,1,20}]  (* A208650 *)
Table[p[n]*n!,{n,1,20}] (* A208651 *)

a(n) = n*Product_{k=1..n-1} binomial(n-1,k). - Jason Yuen, Feb 18 2025

A249346 The exponent of the highest power of 6 dividing the product of the elements on the n-th row of Pascal's triangle.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 4, 0, 0, 10, 10, 4, 13, 8, 3, 0, 6, 0, 28, 20, 12, 24, 15, 6, 20, 10, 0, 16, 47, 22, 26, 0, 30, 48, 33, 18, 73, 56, 39, 40, 42, 24, 47, 28, 9, 54, 57, 16, 62, 40, 18, 46, 23, 0, 82, 32, 84, 94, 87, 44, 92, 52, 36, 0, 102, 72, 107, 76, 45, 82, 50, 18, 128, 94, 60, 100, 65, 30, 72, 36, 0

Offset: 0

Antti Karttunen, Oct 31 2014

Sounds good with MIDI player set to FX-7.

Antti Karttunen, Table of n, a(n) for n = 0..7775

Minimum of terms A187059(n) and A249343(n).

Cf. A001142, A122841, A249151.

a249346 = a122841 . a001142  -- Reinhard Zumkeller, Mar 16 2015

IntegerExponent[#,6]&/@Times@@@Table[Binomial[n,k],{n,0,80},{k,0,n}] (* Harvey P. Dale, Nov 21 2023 *)

A249346(n) = { my(b, s2, s3); s2 = 0; s3 = 0; for(k=0, n, b = binomial(n, k); s2 += valuation(b, 2); s3 += valuation(b, 3)); min(s2,s3); };
for(n=0, 7775, write("b249346.txt", n, " ", A249346(n)));

(define (A249346 n) (min (A187059 n) (A249343 n)))

(define (A249346 n) (A122841 (A001142 n)))

a(n) = min(A187059(n), A249343(n)).
a(n) = A122841(A001142(n)).
Other identities:
a(n) = 0 when A249151(n) < 3.

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A296591 a(n) = Product_{k=0..n} (n + k)!.

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A249433 Integers n such that n! does not divide the product of elements on row n of Pascal's triangle.

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A056606 Squarefree kernel of lcm(binomial(n,0), ..., binomial(n,n)).

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A004788 Number of distinct prime divisors of the numbers in row n of Pascal's triangle.

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A051459 Number of orderings of the subsets of a set with n elements that are compatible with the subsets' sizes; i.e., if A, B are two subsets with A <= B then Card(A) <= Card(B).

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A249345 The exponent of the highest power of 5 dividing the product of the elements on the n-th row of Pascal's triangle.

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A249347 The exponent of the highest power of 7 dividing the product of the elements on the n-th row of Pascal's triangle.

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