A187059 -id:A187059 - 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

A249151 Largest m such that m! divides the product of elements on row n of Pascal's triangle: a(n) = A055881(A001142(n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 2, 4, 10, 7, 12, 6, 4, 1, 16, 2, 18, 4, 6, 10, 22, 11, 4, 12, 2, 6, 28, 25, 30, 1, 10, 16, 6, 36, 36, 18, 12, 40, 40, 6, 42, 10, 23, 22, 46, 19, 6, 4, 16, 12, 52, 2, 10, 35, 18, 28, 58, 47, 60, 30, 63, 1, 12, 10, 66, 16, 22, 49, 70, 41, 72, 36, 4, 18, 10, 12, 78, 80, 2

Offset: 0

Antti Karttunen, Oct 25 2014

A000225 gives the positions of ones.

A006093 seems to give all such k, that a(k) = k.

			              Binomial coeff.   Their product  Largest k!
                 A007318          A001142(n)   which divides
Row 0                1                    1        1!
Row 1              1   1                  1        1!
Row 2            1   2   1                2        2!
Row 3          1   3   3   1              9        1!
Row 4        1   4   6   4   1           96        4! (96 = 4*24)
Row 5      1   5  10  10   5   1       2500        2! (2500 = 1250*2)
Row 6    1   6  15  20  15   6   1   162000        6! (162000 = 225*720)

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms 0..4096 from Antti Karttunen)

One more than A249150.
Cf. A249423 (numbers k such that a(k) = k+1).
Cf. A249429 (numbers k such that a(k) > k).
Cf. A249433 (numbers k such that a(k) < k).
Cf. A249434 (numbers k such that a(k) >= k).
Cf. A249424 (numbers k such that a(k) = (k-1)/2).
Cf. A249428 (and the corresponding values, i.e. numbers n such that A249151(2n+1) = n).
Cf. A249425 (record positions).
Cf. A249427 (record values).
Cf. A001142, A006093, A000225, A007917, A055881, A187059, A249346, A249421, A249430, A249431, A249432.

A249151(n) = { my(uplim,padicvals,b); uplim = (n+3); padicvals = vector(uplim); for(k=0, n, b = binomial(n, k); for(i=1, uplim, padicvals[i] += valuation(b, prime(i)))); k = 1; while(k>0, for(i=1, uplim, if((padicvals[i] -= valuation(k, prime(i))) < 0, return(k-1))); k++); };
\\ Alternative implementation:
A001142(n) = prod(k=1, n, k^((k+k)-1-n));
A055881(n) = { my(i); i=2; while((0 == (n%i)), n = n/i; i++); return(i-1); }
A249151(n) = A055881(A001142(n));
for(n=0, 4096, write("b249151.txt", n, " ", A249151(n)));

from itertools import count
from collections import Counter
from math import comb
from sympy import factorint
def A249151(n):
    p = sum((Counter(factorint(comb(n,i))) for i in range(n+1)),start=Counter())
    for m in count(1):
        f = Counter(factorint(m))
        if not f<=p:
            return m-1
        p -= f # Chai Wah Wu, Aug 19 2025

(define (A249151 n) (A055881 (A001142 n)))

a(n) = A055881(A001142(n)).

A249421 A(n,k) = exponent of the largest power of n-th prime which divides the product of the elements on row (k-1) of Pascal's triangle; a square array read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 5, 2, 0, 0, 0, 2, 1, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 17, 2, 3, 0, 0, 0, 0, 0, 0, 10, 0, 2, 0, 0, 0, 0, 0, 0, 0, 12, 14, 1, 6, 0, 0, 0, 0, 0, 0, 0, 4, 10, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 18, 6, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 13, 6, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 8, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Oct 28 2014

Square array A(n,k), where n = row, k = column, read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ....

A(n,k) is A000040(n)-adic valuation of A001142(k-1).

			The top left corner of the array:
0, 0, 1, 0, 5, 2, 4, 0, 17, 10, 12,  4, 18,  8, 11,  0, 49, 34, 36, 20, 42,
0, 0, 0, 2, 1, 0, 4, 2,  0, 14, 10,  6, 13,  8,  3, 12,  6,  0, 28, 20, 12,
0, 0, 0, 0, 0, 4, 3, 2,  1,  0,  8,  6,  4,  2,  0, 12,  9,  6,  3,  0, 16,
0, 0, 0, 0, 0, 0, 0, 6,  5,  4,  3,  2,  1,  0, 12, 10,  8,  6,  4,  2,  0,
0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0, 10,  9,  8,  7,  6,  5,  4,  3,  2,  1,
0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0, 12, 11, 10,  9,  8,  7,  6,  5,
0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 16, 15, 14, 13,
0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 18, 17,
...

Jeffrey C. Lagarias, Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.

Transpose: A249422.
Row 1: A187059, Row 2: A249343, Row 3: A249345, Row 4 A249347. (Cf. also A249346).
Cf. A000040, A001142, A007318, A249344, A249151.

(define (A249421 n) (A249421bi (A002260 n) (A004736 n)))
(define (A249421bi row col) (A249344bi row (A001142 (- col 1)))) ;; Code for A249344bi given in A249344.

A(n, k) = A249344(n, A001142(k-1)).

A249343 The exponent of the highest power of 3 dividing the product of the elements on the n-th row of Pascal's triangle (A001142(n)).

Original entry on oeis.org

0, 0, 0, 2, 1, 0, 4, 2, 0, 14, 10, 6, 13, 8, 3, 12, 6, 0, 28, 20, 12, 24, 15, 6, 20, 10, 0, 68, 55, 42, 58, 44, 30, 48, 33, 18, 73, 56, 39, 60, 42, 24, 47, 28, 9, 78, 57, 36, 62, 40, 18, 46, 23, 0, 136, 110, 84, 114, 87, 60, 92, 64, 36, 132, 102, 72, 107, 76, 45, 82, 50, 18, 128, 94, 60, 100, 65, 30, 72, 36, 0

Offset: 0

Antti Karttunen, Oct 28 2014

nonn

Paolo Xausa, Table of n, a(n) for n = 0..10000 (terms 0..6560 from Antti Karttunen).
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.

Row sums of A243759.
Row 2 of array A249421.
Cf. A001142, A007949, A187059, A249345, A249347.

a249343 = a007949 . a001142  -- Reinhard Zumkeller, Mar 16 2015

A249343[n_] := Sum[#*((#+1)*3^k - n - 1) & [Floor[n/3^k]], {k, Floor[Log[3, n]]}];
Array[A249343, 100, 0] (* Paolo Xausa, Feb 11 2025 *)

allocatemem(234567890);
A249343(n) = sum(k=0, n, valuation(binomial(n, k), 3));
for(n=0, 6560, write("b249343.txt", n, " ", A249343(n)));

(define (A249343 n) (add A243759 (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) = A007949(A001142(n)).
a(n) = Sum_{k=0..n} A243759(n,k).
a(n) = Sum_{i=1..n} (2*i-n-1)*v_3(i), where v_3(i) = A007949(i) is the exponent of the highest power of 3 dividing i. - Ridouane Oudra, Jun 02 2022
a(n) = Sum_{k=1..floor(log_3(n))} t*((t+1)*3^k - n - 1), where t = floor(n/(3^k)). - Paolo Xausa, Feb 11 2025, derived from Ridouane Oudra's formula above.

A065040 Triangle read by rows: T(m,k) = exponent of the highest power of 2 dividing the binomial coefficient binomial(m,k).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 3, 1, 3, 2, 3, 0, 0, 0, 2, 2, 1, 1, 2, 2, 0, 0, 0, 1, 0, 3, 1, 2, 1, 3, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 2, 1, 2, 0, 3, 2, 3, 0, 2, 1, 2, 0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0

Offset: 0

Claude Lenormand (hlne.lenormand(AT)voono.net), Nov 05 2001

T(m,k) is the number of 'carries' that occur when adding k and m-k in base 2 using the traditional addition algorithm. - Tom Edgar, Jun 10 2014

			Triangle begins:
[0]
[0, 0]
[0, 1, 0]
[0, 0, 0, 0]
[0, 2, 1, 2, 0]
[0, 0, 1, 1, 0, 0]
[0, 1, 0, 2, 0, 1, 0]
[0, 0, 0, 0, 0, 0, 0, 0]
[0, 3, 2, 3, 1, 3, 2, 3, 0]
[0, 0, 2, 2, 1, 1, 2, 2, 0, 0]
[0, 1, 0, 3, 1, 2, 1, 3, 0, 1, 0]
... - _N. J. A. Sloane_, Aug 21 2021

Antti Karttunen, Table of n, a(n) for n = 0..10010; the first 141 antidiagonals, flattened
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.

Row sums: A187059.

Cf. A007318, A007814, A001511, A000120, A047999, A049606, A000680, A048881, A011371, A005187, A000265, A001316, A001317, A243759.

A065040 := (n, k) -> padic[ordp](binomial(n, k), 2):
seq(seq(A065040(n,k), k=0..n), n=0..13); # Peter Luschny, Aug 15 2017

T[m_, k_] := IntegerExponent[Binomial[m, k], 2]; Table[T[m, k], {m, 0, 13}, {k, 0, m}] // Flatten (* Jean-François Alcover, Oct 06 2016 *)

T(m,k)=hammingweight(k)+hammingweight(m-k)-hammingweight(m)
for(m=0,9,for(k=0,m,print1(T(m,k)", "))) \\ Charles R Greathouse IV, Mar 26 2013

As an array f(i,j) = f(j,i) = T(i+j,j) read by antidiagonals: f(0,j) = 0, f(1,j) = A007814(j+1), f(i,j) = Sum_{k=0..i-1} (f(1,j+k) - f(1,k)). [corrected by Kevin Ryde, Oct 07 2021]
The n-th term a(n) is equal to the binomial coefficient binomial(m,k), where m = floor((1+sqrt(8*n+1))/2) - 1 and k = n - m(m+1)/2. Also a(n) = g(m) - g(k) - g(m-k), where g(x) = Sum_{i=1..floor(log_2(x))} floor(x/2^i), m = floor((1+sqrt(8*n+1))/2) - 1, k = n - m(m+1)/2. - Hieronymus Fischer, May 05 2007
T(m,k) <= log_2 m, for m > 0. - Charles R Greathouse IV, Mar 26 2013
T(m,k) = log_2(A082907(m,k)). - Tom Edgar, Jun 10 2014
From Antti Karttunen, Oct 28 2014: (Start)
a(n) = A007814(A007318(n)).
a(n) * A047999(n) = 0 and a(n) + A047999(n) > 0 for all n.
(End)

Name clarified by Antti Karttunen, Oct 28 2014

A249150 Number of trailing zeros in the factorial base representation of products of binomial coefficients: a(n) = A230403(A001142(n)).

Original entry on oeis.org

0, 0, 1, 0, 3, 1, 5, 0, 1, 3, 9, 6, 11, 5, 3, 0, 15, 1, 17, 3, 5, 9, 21, 10, 3, 11, 1, 5, 27, 24, 29, 0, 9, 15, 5, 35, 35, 17, 11, 39, 39, 5, 41, 9, 22, 21, 45, 18, 5, 3, 15, 11, 51, 1, 9, 34, 17, 27, 57, 46, 59, 29, 62, 0, 11, 9, 65, 15, 21, 48, 69, 40, 71, 35, 3, 17, 9, 11, 77, 79, 1

Offset: 0

Antti Karttunen, Oct 25 2014

nonn

a(n) = A249151(n)-1. Please see the comments and graph of that sequence.

One less than A249151.
Cf. A249423 (values k such that a(k) = k).
Cf. A249425 (record positions).
Cf. A249426 (record values).
Cf. A001142, A187059, A230403.

(define (A249150 n) (A230403 (A001142 n)))

a(n) = A230403(A001142(n)).

A174605 Partial sums of A011371.

Original entry on oeis.org

0, 0, 1, 2, 5, 8, 12, 16, 23, 30, 38, 46, 56, 66, 77, 88, 103, 118, 134, 150, 168, 186, 205, 224, 246, 268, 291, 314, 339, 364, 390, 416, 447, 478, 510, 542, 576, 610, 645, 680, 718, 756, 795, 834, 875, 916, 958, 1000, 1046, 1092, 1139, 1186, 1235, 1284, 1334

Offset: 0

Jonathan Vos Post, Mar 23 2010

Exponent of 2 in the superfactorials, i.e., a(n) = A007814(A000178(n)). - Ralf Stephan, Jan 03 2014

Amiram Eldar, Table of n, a(n) for n = 0..10000
Hsien-Kuei Hwang, Svante Janson and Tsung-Hsi Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47. Also first authors' copy, 2016.
Jeffrey C. Lagarias and Harsh Mehta, Products of Binomial Coefficients and Unreduced Farey Fractions, arXiv:1409.4145 [math.NT], 2014, see a(n) = ord_p(N*_n) for p=2 in theorem 4.1.
A. Mir, F. Rossello and L. Rotger, A new balance index for phylogenetic trees, arXiv preprint arXiv:1202.1223 [q-bio.PE], 2012, see proposition 15, a(n) = x(n) = f(n+1).

Cf. A000120, A011371 (first differences).
Cf. A000178 (superfactorials), A007814 (2-adic valuation), A272011 (binary exponents).
Cf. A249152 (hyperfactorial valuation), A187059 (binomial valuation), A173345 (superfactorial 10-valuation).

a:= proc(n) option remember; `if`(n<1, 0,
      a(n-1)+n-add(i, i=Bits[Split](n)))
    end:
seq(a(n), n=0..54);  # Alois P. Heinz, Oct 30 2021

Accumulate[Table[n-DigitCount[n,2,1],{n,0,130}]] (* Harvey P. Dale, Feb 26 2015 *)
a[n_] := IntegerExponent[BarnesG[n + 2], 2]; Array[a, 100, 0] (* Amiram Eldar, Aug 08 2024 *)

a(n) = n++; my(v=binary(n),t=#v-1); for(i=1,#v, if(v[i],v[i]=t++,t--)); (n^2 - fromdigits(v,2))>>1; \\ Kevin Ryde, Oct 29 2021

def A174605(n): return (n*(n+1)>>1)-(n+1)*n.bit_count()-(sum((m:=1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1))>>1)  # Chai Wah Wu, Nov 12 2024

a(n) = Sum_{i=0..n} A011371(i).
From Kevin Ryde, Oct 29 2021: (Start)
a(n) = n*(n+1)/2 - A000788(n).
a(n) ~ (n^2)/2 + O(n*log_2(n)). [Lagarias and Mehta, theorem 4.2 with p=2]
a(n) = ( (n+1)^2 - Sum_{i=1..k} (e[i]+2*i-1) * 2^e[i] )/2, where binary expansion n+1 = 2^e[1] + ... + 2^e[k] with descending exponents e[1] > e[2] > ... > e[k] (A272011).
(End)

A249152 Exponent of 2 in the hyperfactorials: a(n) = A007814(A002109(n)).

Original entry on oeis.org

0, 0, 2, 2, 10, 10, 16, 16, 40, 40, 50, 50, 74, 74, 88, 88, 152, 152, 170, 170, 210, 210, 232, 232, 304, 304, 330, 330, 386, 386, 416, 416, 576, 576, 610, 610, 682, 682, 720, 720, 840, 840, 882, 882, 970, 970, 1016, 1016, 1208, 1208, 1258, 1258, 1362, 1362, 1416, 1416, 1584, 1584, 1642, 1642

Offset: 0

Antti Karttunen, Oct 25 2014

nonn

This is the function ord_2(D*_n) listed in the leftmost column of Table 7.1 in Lagarias & Mehta 2014 paper (on page 19).

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.
Luca Onnis, On the p-adic valuation of a hyperfactorial, arXiv:2109.05616 [math.NT], 2021.

Bisection: A249153.
Cf. A174605, A187059, A002109, A007814, A091512, A143157.
Cf. A133457 (binary exponents).

[0] cat [&+[i*Valuation(i, 2):i in [1..n]]:n in [1..60]]; // Marius A. Burtea, Oct 18 2019

with(padic): seq(add(i*ordp(i, 2), i=1..n), n=0..60); # Ridouane Oudra, Oct 17 2019

Table[i=0;Hyperfactorial@n//.x_/;EvenQ@x:>(i++;x/2);i,{n,0,60}] (* Giorgos Kalogeropoulos, Oct 28 2021 *)

a(n) = sum(i=1, n, i*valuation(i, 2)); \\ Michel Marcus, Sep 14 2021

a(n) = my(v=binary(n),t=0); forstep(j=#v,1,-1, if(v[j],v[j]=t--,t++)); (n^2 + fromdigits(v,2))>>1; \\ Kevin Ryde, Nov 03 2021

def A249152(n): return sum(i*(~i&i-1).bit_length() for i in range(2,n+1,2)) # Chai Wah Wu, Jul 11 2022

a(n) = 2 * A143157(floor(n/2)).
a(n) = A174605(n) + A187059(n). [Lagarias and Mehta theorem 4.1 for p=2]
a(n) = Sum_{i=1..n} i*v_2(i), where v_2(i) = A007814(i) is the exponent of the highest power of 2 dividing i. - Ridouane Oudra, Oct 17 2019
a(n) ~ (n^2+2n)/2 as n -> infinity. - Luca Onnis, Oct 17 2021
a(n) ~ ((A011371(n))^2)/2 as n -> infinity. - Luca Onnis, Nov 02 2021
From Kevin Ryde, Nov 03 2021: (Start)
a(2n) = a(2n+1) = 2*a(n) + n*(n+1).
a(n) = ( n^2 + Sum_{j=1..k} (e[j]-2*j+1) * 2^e[j] )/2, where binary expansion n = 2^e[1] + ... + 2^e[k] with ascending exponents e[1] < e[2] < ... < e[k] (A133457).
(End)
a(n) = Sum_{j=1..floor(log_2(n))} j*2^j*round(n/2^(j+1))^2, for n>=1. - Ridouane Oudra, Oct 01 2022

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.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

Maxima

PARI

PARI

Python

Sage

Scheme

Formula

Extensions

A249151 Largest m such that m! divides the product of elements on row n of Pascal's triangle: a(n) = A055881(A001142(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Scheme

Formula

A249421 A(n,k) = exponent of the largest power of n-th prime which divides the product of the elements on row (k-1) of Pascal's triangle; a square array read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula

A249343 The exponent of the highest power of 3 dividing the product of the elements on the n-th row of Pascal's triangle (A001142(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Scheme

Formula

A065040 Triangle read by rows: T(m,k) = exponent of the highest power of 2 dividing the binomial coefficient binomial(m,k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A249150 Number of trailing zeros in the factorial base representation of products of binomial coefficients: a(n) = A230403(A001142(n)).

Views

Author

Keywords

Comments

Crossrefs

Programs