A256113 -id:A256113 - cp's OEIS frontend

A001142 a(n) = Product_{k=1..n} k^(2k - 1 - n).

1, 1, 2, 9, 96, 2500, 162000, 26471025, 11014635520, 11759522374656, 32406091200000000, 231627686043080250000, 4311500661703860387840000, 209706417310526095716965894400, 26729809777664965932590782608648192

Offset: 0

N. J. A. Sloane

Absolute value of determinant of triangular matrix containing binomial coefficients.
These are also the products of consecutive horizontal rows of the Pascal triangle. - Jeremy Hehn (ROBO_HEN5000(AT)rose.net), Mar 29 2007
Limit_{n->oo} a(n)*a(n+2)/a(n+1)^2 = e, as follows from lim_{n->oo} (1 + 1/n)^n = e. - Harlan J. Brothers, Nov 26 2009
A000225 gives the positions of odd terms. - Antti Karttunen, Nov 02 2014
Product of all unreduced fractions h/k with 1 <= k <= h <= n. - Jonathan Sondow, Aug 06 2015
a(n) is a product of the binomial coefficients from the n-th row of the Pascal triangle, for n= 0, 1, 2, ... For n > 0, a(n) means the number of all maximum chains in the poset formed by the n-dimensional Boolean cube {0,1}^n and the relation "precedes by weight". This relation is defined over {0,1}^n as follows: for arbitrary vectors u, v of {0,1}^n we say that "u precedes by weight v" if wt(u) < wt(v) or if u = v, where wt(u) denotes the (Hamming) weight of u. For more details, see the sequence A051459. - Valentin Bakoev, May 17 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..50
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537. Zbl 1133.11012.
Harlan J. Brothers, Finding e in Pascal's Triangle, Mathematics Magazine, 85 (2012), p. 51.
Harlan J. Brothers, Pascal's Triangle: The Hidden Stor-e, The Mathematical Gazette, 96 (2012), 145-148.
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.
Leroy Quet, Problem 1636, Mathematics Magazine, Dec. 2001, p. 403.

Cf. A000178, A002109, A007318, A000225, A056077, A249421, A187059 (2-adic valuation), A249343, A249345, A249346, A249347, A249151.

Cf. also A004788, A056606 (squarefree kernel), A256113.

List([0..15], n-> Product([0..n], k-> Binomial(n,k) )); # G. C. Greubel, May 23 2019

a001142 = product . a007318_row -- Reinhard Zumkeller, Mar 16 2015

[(&*[Binomial(n,k): k in [0..n]]): n in [0..15]]; // G. C. Greubel, May 23 2019

a:=n->mul(binomial(n,k), k=0..n): seq(a(n), n=0..14); # Zerinvary Lajos, Jan 22 2008

Table[Product[k^(2*k - 1 - n), {k, n}], {n, 0, 20}] (* Harlan J. Brothers, Nov 26 2009 *)
Table[Hyperfactorial[n]/BarnesG[n+2], {n, 0, 20}] (* Peter Luschny, Nov 29 2015 *)
Table[Product[(n - k + 1)^(n - 2 k + 1), {k, 1, n}], {n, 0, 20}] (* Harlan J. Brothers, Aug 26 2023 *)

a(n):= prod(binomial(n,k),k,0,n); n : 15; for i from 0 thru n do print (a(i)); /* Valentin Bakoev, May 17 2019 */

for(n=0,16,print(prod(m=1,n,binomial(n,m))))

A001142(n) = prod(k=1, n, k^((k+k)-1-n)); \\ Antti Karttunen, Nov 02 2014

from math import factorial, prod
from fractions import Fraction
def A001142(n): return prod(Fraction((k+1)**k,factorial(k)) for k in range(1,n)) # Chai Wah Wu, Jul 15 2022

a = lambda n: prod(k^k/factorial(k) for k in (1..n))
[a(n) for n in range(20)] # Peter Luschny, Nov 29 2015

(define (A001142 n) (mul (lambda (k) (expt k (+ k k -1 (- n)))) 1 n))
(define (mul intfun lowlim uplim) (let multloop ((i lowlim) (res 1)) (cond ((> i uplim) res) (else (multloop (+ 1 i) (* res (intfun i)))))))
;; Antti Karttunen, Oct 28 2014

a(n) = C(n, 0)*C(n, 1)* ... *C(n, n).
From Harlan J. Brothers, Nov 26 2009: (Start)
a(n) = Product_{j=1..n-2} Product_{k=1..j} (1+1/k)^k, n >= 3.
a(1) = a(2) = 1, a(n) = a(n-1) * Product_{k=1..n-2} (1+1/k)^k. (End)
a(n) = hyperfactorial(n)/superfactorial(n) =  A002109(n)/A000178(n). - Peter Luschny, Jun 24 2012
a(n) ~ A^2 * exp(n^2/2 + n - 1/12) / (n^(n/2 + 1/3) * (2*Pi)^((n+1)/2)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015
a(n) = Product_{i=1..n} Product_{j=1..i} (i/j). - Pedro Caceres, Apr 06 2019
a(n) = Product_{k=1..n} (n - k + 1)^(n - 2*k + 1). - Harlan J. Brothers, Aug 26 2023

More terms from James Sellers, May 01 2000

Better description from Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001

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

A004789 Least k such that number of distinct prime divisors of the numbers in row k of Pascal's triangle is n.

Original entry on oeis.org

0, 2, 4, 6, 10, 11, 16, 18, 22, 23, 29, 35, 39, 42, 44, 47, 55, 59, 62, 69, 71, 78, 79, 83, 89, 100, 102, 104, 107, 111, 119, 130, 131, 138, 139, 149, 153, 159, 164, 167, 174, 179, 181, 191, 194, 197, 199, 215, 223, 228, 230, 233, 239, 250, 251, 259, 263, 269, 272, 279, 282

Offset: 0

Clark Kimberling

nonn

A004788(a(n)) = n and A004788(m) != n for m < a(n). - Reinhard Zumkeller, Mar 15 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..250

Cf. A004788, A256113.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a004789 = fromJust . (`elemIndex` a004788_list)
-- Reinhard Zumkeller, Mar 15 2015

a(n) = {irow = 0; while(omega(prod(i=0, irow, binomial(irow, i)))!=n, irow++); return (irow);} \\ Michel Marcus, May 13 2013

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A001142 a(n) = Product_{k=1..n} k^(2k - 1 - n).

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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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A004789 Least k such that number of distinct prime divisors of the numbers in row k of Pascal's triangle is n.

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