A249151 -id:A249151 - cp's OEIS frontend

A249424 Odd integers n such that A249151(n) = (n-1)/2.

3, 5, 9, 13, 21, 23, 25, 33, 37, 45, 57, 61, 73, 81, 85, 93, 105, 117, 121, 133, 141, 145, 157, 165, 177, 193, 201, 205, 213, 217, 225, 253, 261, 273, 277, 297, 301, 313, 325, 333, 345, 357, 361, 381, 385, 393, 397, 421, 445, 453, 457, 465, 477, 481, 501, 513, 525, 537, 541, 553, 561, 565, 585, 613, 621, 625, 633, 661, 673, 693, 697

Offset: 1

Antti Karttunen, Oct 28 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..310

A249428 gives the corresponding values (n-1)/2.
Subsequence of A249433.
Cf. A249151, A249423, A249425, A249427.

A249425 Positions of records in A249150 and A249151.

Original entry on oeis.org

0, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 35, 39, 42, 46, 52, 58, 60, 62, 66, 70, 72, 78, 79, 82, 83, 88, 89, 96, 100, 102, 104, 106, 107, 112, 125, 126, 130, 131, 136, 138, 143, 148, 149, 153, 156, 159, 162, 164, 166, 167, 172, 174, 179, 181, 190, 192, 194, 196, 197, 199, 207, 209, 219, 222, 226, 228, 232, 238, 240, 250, 256

Offset: 1

Antti Karttunen, Oct 28 2014

nonn

This is not a subsequence of A249434, as k = a(36) = 125 is the first case where A249431(k) < 0 as A249151(125) = 119.

Antti Karttunen, Table of n, a(n) for n = 1..999

Cf. A249150, A249151, A249431, A249434.

A249431 a(n) = A249151(n) - n.

Original entry on oeis.org

1, 0, 0, -2, 0, -3, 0, -6, -6, -5, 0, -4, 0, -7, -10, -14, 0, -15, 0, -15, -14, -11, 0, -12, -20, -13, -24, -21, 0, -4, 0, -30, -22, -17, -28, 1, 0, -19, -26, 1, 0, -35, 0, -33, -21, -23, 0, -28, -42, -45, -34, -39, 0, -51, -44, -20, -38, -29, 0, -12, 0, -31, 1, -62, -52, -55, 0, -51, -46, -20, 0, -30, 0

Offset: 0

Antti Karttunen, Nov 02 2014

It seems that A006093 gives the positions of zeros.

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

A249433 and A249434 give the positions of negative and nonnegative values, respectively.
A249430 gives the position where +n appears for the first time, A249432 the positions of records of positive values.
Cf. A006093, A249151.

from itertools import count
from collections import Counter
from math import comb
from sympy import factorint
def A249431(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-n
        p -= f # Chai Wah Wu, Aug 19 2025

(definec (A249431 n) (- (A249151 n) n))

a(n) = A249151(n) - n.

A249423 Integers n such that A249150(n) = n; integers n such that A249151(n) = n+1.

Original entry on oeis.org

0, 35, 39, 62, 79, 83, 89, 104, 107, 131, 143, 149, 153, 159, 164, 167, 175, 179, 181, 194, 197, 199, 207, 209, 219, 259, 263, 269, 272, 274, 279, 285, 287, 296, 299, 305, 307, 311, 314, 319, 329, 339, 356, 359, 363, 373, 377, 379, 384, 389, 391, 395, 399, 407, 415, 417, 419, 424, 428, 431, 441, 449, 455, 461, 467, 475, 489, 512

Offset: 1

Antti Karttunen, Oct 28 2014

nonn

Integers n such that {product of elements on row n of Pascal's triangle} is divisible by (n+1)! but not by (n+2)!

Antti Karttunen, Table of n, a(n) for n = 1..732

Subsequence of A249434 and of A249429; it differs from the latter for the first time at n=17, where a(17) = 175 > 174 = A249429(17).

Cf. A000142, A001142, A007318, A249150, A249151, A249424, A249425, A249426.

A249427 Record values in A249151.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 63, 66, 70, 72, 78, 80, 82, 84, 88, 90, 96, 100, 102, 105, 106, 108, 112, 119, 126, 130, 132, 136, 138, 144, 148, 150, 154, 156, 160, 162, 165, 166, 168, 172, 178, 180, 182, 190, 192, 195, 196, 198, 200, 208, 210, 220, 222, 226, 228, 232, 238, 240, 250, 256

Offset: 1

Antti Karttunen, Oct 28 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..999

One more than A249426.
Cf. A249151, A249425.
Differs from A006093 for the first at n=19, where a(19) = 63, while A006093(19) = 66.

(define (A249427 n) (A249151 (A249425 n)))

a(n) = A249151(A249425(n)).

a(n) = A249426(n) + 1.

A249436 Integers n such that n/2 < A249151(n) < n.

Original entry on oeis.org

11, 29, 44, 55, 59, 69, 71, 111, 119, 125, 139, 188, 215, 223, 230, 233, 239, 251, 324, 335, 349, 351, 447, 458, 474, 479, 493, 494, 503, 509, 560, 593, 599, 647, 662, 701, 714, 719, 831, 835, 849, 895, 956, 959, 979, 991, 1000, 1007, 1019, 1034, 1063, 1077, 1169, 1224, 1319, 1322, 1364, 1376, 1424, 1427, 1448, 1507

Offset: 1

Antti Karttunen, Nov 02 2014

nonn

Integers n such that A001142(n) [product of elements on row n of Pascal's triangle] is divisible by floor[(n+1)/2]! but not by n!

These are the abscissas of "stray points" in the sparsely populated region between the two topmost rays visible in the scatter plot of A249151 which have slopes 1 and 1/2: A249434 (A006093) and A249424.

Antti Karttunen, Table of n, a(n) for n = 1..115

A249437 gives the corresponding values at those points.
Subsequence of A249433.
Cf. A000142, A001142, A249151, A006093, A249424, A249434.

A249428 Numbers n such that A249151(2n+1) = n.

Original entry on oeis.org

1, 2, 4, 6, 10, 11, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270, 276, 280, 282, 292, 306, 310, 312, 316, 330, 336, 346, 348

Offset: 1

Antti Karttunen, Oct 28 2014

nonn

Seems to be A006093 with at least one additional term, 11 at a(6).

Antti Karttunen, Table of n, a(n) for n = 1..310

Cf. A006093 (seems to be a subsequence).

Cf. also A056077, A249151, A249424.

(define (A249428 n) (A249151 (A249424 n)))

a(n) = A249151(A249424(n)).

A249437 a(n) = A249151(A249436(n)).

Original entry on oeis.org

7, 25, 23, 35, 47, 49, 41, 67, 89, 119, 97, 139, 113, 131, 120, 188, 181, 233, 168, 265, 209, 241, 259, 329, 309, 357, 337, 449, 463, 501, 288, 461, 409, 329, 493, 548, 571, 656, 649, 681, 577, 515, 791, 709, 601, 837, 805, 919, 995, 528, 721, 901, 961, 1009, 1121, 895, 1273, 977, 1063, 1121, 1376, 1273, 1057, 840, 1189, 1009

Offset: 1

Antti Karttunen, Nov 02 2014

nonn

These are the ordinates of "stray points", listed in the order they appear (from left to right) in the scatter plot of A249151, in the sparsely populated region between the two topmost rays which have slopes 1 and 1/2: A249434 (A006093) and A249424.

Antti Karttunen, Table of n, a(n) for n = 1..115

A249438 gives the same sequence sorted into ascending order.

Cf. A249436, A249151, A006093, A249424, A249434.

(define (A249437 n) (A249151 (A249436 n)))

a(n) = A249151(A249436(n)).

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

Original entry on oeis.org

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

A187059 The exponent of highest power of 2 dividing the product of the elements of the n-th row of Pascal's triangle (A001142).

Original entry on oeis.org

0, 0, 1, 0, 5, 2, 4, 0, 17, 10, 12, 4, 18, 8, 11, 0, 49, 34, 36, 20, 42, 24, 27, 8, 58, 36, 39, 16, 47, 22, 26, 0, 129, 98, 100, 68, 106, 72, 75, 40, 122, 84, 87, 48, 95, 54, 58, 16, 162, 116, 119, 72, 127, 78, 82, 32, 147, 94, 98, 44, 108, 52, 57, 0, 321, 258, 260, 196, 266, 200, 203, 136, 282, 212, 215, 144, 223, 150, 154, 80, 322, 244, 247, 168, 255, 174, 178, 96, 275, 190, 194, 108, 204, 116, 121, 32, 418, 324, 327, 232, 335

Offset: 0

Bruce Reznick, Mar 05 2011

The exponent of the highest power of 2 which divides Product_{k=0..n} binomial(n, k). This can be computed using de Polignac's formula.

This is the function ord_2(Ḡ_n) extensively studied in Lagarias-Mehta (2014), and plotted in Fig. 1.1. - Antti Karttunen, Oct 22 2014

			For example, if n = 4, the power of 2 that divides 1*4*6*4*1 is 5.

I. Niven, H. S. Zuckerman, H. L. Montgomery, An Introduction to the Theory of Numbers, Wiley, 1991, pages 182, 183, 187 (Ex. 34).

Antti Karttunen and Paul Tek, Table of n, a(n) for n = 0..8191 (First 4096 terms from Karttunen)
J. Lagarias, Products of binomial coefficients and Farey fractions, Lecture in DIMACS Conference on Challenges of Identifying Integer Sequences, October 9, 2014; Part 1, Part 2.
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.

Row sums of triangular table A065040.
Row 1 of array A249421.
Cf. A000295 (a(2^k-2)), A000337 (a(2^k)), A005803 (a(2^k-3)), A036799 (a(2^k+1)), A109363 (a(2^k-4)).
Cf. A000178, A000788, A001142, A002109, A007318, A007814, A174605, A249152, A249150, A249151, A249154, A249343, A249345, A249346, A249347.

a187059 = a007814 . a001142  -- Reinhard Zumkeller, Mar 16 2015

a[n_] := Sum[IntegerExponent[Binomial[n, k], 2], {k, 0, n}]; Array[a, 100, 0]
A187059[n_] := Sum[#*((#+1)*2^k - n - 1) & [Floor[n/2^k]], {k, Floor[Log2[n]]}];
Array[A187059, 100, 0] (* Paolo Xausa, Feb 11 2025 *)
2*Accumulate[#] - Range[Length[#]]*# & [DigitCount[Range[0, 99], 2, 1]] (* Paolo Xausa, Feb 11 2025 *)

a(n)=sum(k=0,n,valuation(binomial(n,k),2))

\\ Much faster version, based on code for A065040 by Charles R Greathouse IV which if reduced even further gives the formula a(n) = 2*A000788(n) - A249154(n):
A065040(m,k) = (hammingweight(k)+hammingweight(m-k)-hammingweight(m));
A187059(n) = sum(k=0, n, A065040(n, k));
for(n=0, 4095, write("b187059.txt", n, " ", A187059(n)));
\\ Antti Karttunen, Oct 25 2014

def A187059(n): return (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)) # Chai Wah Wu, Nov 11 2024

a(2^k-1) = 0 (19th century); a(2^k) = (k-1)*2^k+1 for k >= 1. (Use de Polignac.)
a(n) = Sum_{i=0..n} A065040(n,i) [where the entries of triangular table A065040(m,k) give the exponent of the maximal power of 2 dividing binomial coefficient A007318(m,k)].
a(n) = A007814(A001142(n)). - Jason Kimberley, Nov 02 2011
a(n) = A249152(n) - A174605(n). [Exponent of 2 in the n-th hyperfactorial minus exponent of 2 in the n-th superfactorial. Cf. for example Lagarias & Mehta paper or Peter Luschny's formula for A001142.] - Antti Karttunen, Oct 25 2014
a(n) = 2*A000788(n) - A249154(n). - Antti Karttunen, Nov 02 2014
a(n) = Sum_{i=1..n} (2*i-n-1)*v_2(i), where v_2(i) = A007814(i) is the exponent of the highest power of 2 dividing i. - Ridouane Oudra, Jun 02 2022
a(n) = Sum_{k=1..floor(log_2(n))} t*((t+1)*2^k - n - 1), where t = floor(n/(2^k)). - Paolo Xausa, Feb 11 2025, derived from Ridouane Oudra's formula above.

Name clarified by Antti Karttunen, Oct 22 2014

cp's OEIS Frontend

A249424 Odd integers n such that A249151(n) = (n-1)/2.

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A249425 Positions of records in A249150 and A249151.

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A249431 a(n) = A249151(n) - n.

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A249423 Integers n such that A249150(n) = n; integers n such that A249151(n) = n+1.

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A249427 Record values in A249151.

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A249436 Integers n such that n/2 < A249151(n) < n.

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A249428 Numbers n such that A249151(2n+1) = n.

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A249437 a(n) = A249151(A249436(n)).

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

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