A004788 -id:A004788 - 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

A256113 Table read by rows: T(1,1) = 1, for n > 1: row n = union of distinct prime factors occurring in terms of n-th row of Pascal's triangle, cf. A007318.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 5, 2, 3, 5, 3, 5, 7, 2, 5, 7, 2, 3, 7, 2, 3, 5, 7, 2, 3, 5, 7, 11, 2, 3, 5, 7, 11, 2, 3, 5, 11, 13, 2, 3, 7, 11, 13, 3, 5, 7, 11, 13, 2, 3, 5, 7, 11, 13, 2, 5, 7, 11, 13, 17, 2, 3, 5, 7, 11, 13, 17, 2, 3, 7, 11, 13, 17, 19, 2, 3, 5, 11, 13

Offset: 1

Reinhard Zumkeller, Mar 16 2015

			.  n |   T(n,k)   |                A001142(n) | A007318(n,0..n)
. ---+------------+---------------------------+-------------------------
.  1 | 1          |                         1 | 1  1
.  2 | 2          |                         2 | 1  2  1
.  3 | 3          |                         9 | 1  3  3   1
.  4 | 2 3        |                        96 | 1  4  6   4   1
.  5 | 2 5        |                      2500 | 1  5 10  10   5   1
.  6 | 2 3 5      |                    162000 | 1  6 15  20  15   6   1
.  7 | 3 5 7      |                  26471025 | 1  7 21  35  35  21   7   1
.  8 | 2 5 7      |               11014635520 | 1  8 28  56  70  56  28 ...
.  9 | 2 3 7      |            11759522374656 | 1  9 36  84 126 126  84 ...
. 10 | 2 3 5 7    |         32406091200000000 | 1 10 45 120 210 252 210 ...
. 11 | 2 3 5 7 11 |     231627686043080250000 | 1 11 55 165 330 462 462 ...
. 12 | 2 3 5 7 11 | 4311500661703860387840000 | 1 12 66 220 495 792 924 ...

Reinhard Zumkeller, b-file >>Rows n = 1..1000 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A027748, A001142, A004788 (row lengths), A056606 (row products).

a256113 n k = a256113_tabf !! (n-1) !! (n-1)
a256113_row n = a256113_tabf !! (n-1)
a256113_tabf = map a027748_row $ tail a001142_list

A158973 a(n) = count of numbers k <= n such that all proper divisors of k are divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 5, 7, 6, 7, 6, 11, 7, 9, 9, 10, 8, 12, 9, 13, 11, 11, 10, 17, 11, 12, 12, 14, 11, 18, 12, 16, 14, 14, 14, 20, 13, 15, 15, 20, 14, 20, 15, 19, 20, 17, 16, 25, 17, 20, 18, 20, 17, 23, 19, 24, 19, 19, 18, 29, 19, 21, 24, 24, 21, 25, 20, 24, 22, 27, 21, 32, 22, 24, 26

Offset: 1

Jaroslav Krizek, Apr 01 2009

nonn

For primes p, a(p) = A036234(p) = A000720(p) + 1.

			For n = 8 we have the following proper divisors for k <= n: {1}, {1}, {1}, {1, 2}, {1}, {1, 2, 3}, {1}, {1, 2, 4}. Only k = 6 has a proper divisor that is not a divisor of 8, viz. 3. Hence a(8) = 7.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A000720, A036234.

[ #[ k: k in [1..n] | forall(t){ d: d in Divisors(k) | d eq k or d in Divisors(n) } ]: n in [1..75] ];

N:= 1000: # to get a(1) to a(N)
A:= Vector(N, numtheory:-tau):
for p in select(isprime,[2,seq(i,i=3..N,2)]) do
for d from 0 to floor(log[p](N))-1 do
  R:= [seq(seq(p^d*(i+p*j), j=1..floor((N/p^d - i)/p)), i=1..p-1)];
  A[R]:= map(`+`,A[R],1);
od
od:
convert(A,list); # Robert Israel, Nov 24 2015

f[n_] := Block[{d = Most@ Divisors@ n}, Select[Range@ n, Union[Most@ Divisors@ #, d] == d &]]; Array[Length@ f@ # &, {75}] (* Michael De Vlieger, Nov 25 2015 *)

a004788(n) = {sfp = Set(); for (k=1, n-1, sfp = setunion(sfp, Set(factor(binomial(n, k))[, 1]))); return (length(sfp));}
a(n) = numdiv(n) + a004788(n-1); \\ Altug Alkan, Nov 25 2015

a(n) = A000005(n) + A004788(n-1). - Vladeta Jovovic, Apr 07 2009 (Corrected by Altug Alkan, Nov 25 2015)

Edited and extended by Klaus Brockhaus, Apr 06 2009

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

A323444 Sum of exponents in prime-power factorization of Product_{k=0..n} binomial(n,k) (A001142).

Original entry on oeis.org

0, 0, 1, 2, 6, 6, 11, 10, 23, 28, 33, 28, 45, 38, 44, 50, 86, 74, 96, 82, 106, 110, 114, 96, 147, 150, 153, 182, 211, 184, 215, 186, 281, 280, 279, 278, 347, 308, 306, 304, 380, 336, 374, 328, 368, 408, 403, 352, 489, 482, 524, 516, 559, 498, 596, 586, 686, 674

Offset: 0

Daniel Suteu, Jan 15 2019

nonn

Also sum of exponents in prime-power factorization of hyperfactorial(n) / superfactorial(n).

			a(4) = 6 because C(4,0)*C(4,1)*C(4,2)*C(4,3)*C(4,4) = 2^5 * 3^1 and 5 + 1 = 6, where C(n,k) is the binomial coefficient.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, International Journal of Number Theory, Vol. 12, No. 1 (2016), pp. 57-91; arXiv preprint, arXiv:1409.4145 [math.NT], 2014-2015.
Eric Weisstein's World of Mathematics, Hyperfactorial.
Eric Weisstein's World of Mathematics, Superfactorial.
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences related to factorial numbers.

Cf. A000178, A001142, A001222, A002109, A004788, A007318, A022559, A303279, A303281.

Array[Sum[PrimeOmega@ Binomial[#, k], {k, 0, #}] &, 57] (* Michael De Vlieger, Jan 19 2019 *)

a(n) = sum(k=0, n, bigomega(binomial(n, k)));

a(n) = my(t=0); sum(k=1, n, my(b=bigomega(k)); t+=b; k*b-t);

first(n) = my(res = List([0]), r=0, t=0, b=0); for(k=1, n, b=bigomega(k); t += b; r += k*b-t; listput(res, r)); res \\ adapted from Daniel Suteu \\ David A. Corneth, Jan 16 2019

a(n) = A303281(n) - A303279(n), for n > 0.

a(n) = A001222(A001142(n)).

cp's OEIS Frontend

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

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A256113 Table read by rows: T(1,1) = 1, for n > 1: row n = union of distinct prime factors occurring in terms of n-th row of Pascal's triangle, cf. A007318.

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A158973 a(n) = count of numbers k <= n such that all proper divisors of k are divisors of n.

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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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Haskell

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A323444 Sum of exponents in prime-power factorization of Product_{k=0..n} binomial(n,k) (A001142).

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