A001142 -id:A001142 - cp's OEIS frontend

A132858 Composite "antimutinous" numbers. An antimutinous number is an integer m > 1 where m/p^k < p, where p is the largest prime divisor of m and p^k is the largest power of p dividing m.

4, 6, 8, 9, 10, 14, 15, 16, 18, 20, 21, 22, 25, 26, 27, 28, 32, 33, 34, 35, 38, 39, 42, 44, 46, 49, 50, 51, 52, 54, 55, 57, 58, 62, 64, 65, 66, 68, 69, 74, 75, 76, 77, 78, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 106, 110, 111, 114, 115, 116, 117, 118

Offset: 1

Leroy Quet, Nov 21 2007

nonn

{a(k)-1} is the complement of sequence A056077. In other words, {a(k)} contains precisely those positive integers m where A001142(m-1) (= product{k=1 to m-1} k^(2k-m)) is not divisible by all primes <= m-1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A027855, A132982, A027854, A056077, A001142.

antiQ[n_] := Module[{f = FactorInteger[n], p, k}, p = f[[-1, 1]]; k = f[[-1, 2]]; n/p^k < p]; Select[Range[118], CompositeQ[#] && antiQ[#] &] (* Amiram Eldar, Feb 24 2020 *)

Extended by Ray Chandler, Nov 17 2008

A260610 Suprafactorials: Product of first n hyperfactorials divided by the product of the first n superfactorials.

Original entry on oeis.org

1, 1, 2, 18, 1728, 4320000, 699840000000, 18525482136000000000, 204051433560311070720000000000, 2399547398533110254947300351672320000000000, 77759951835586717141477466390085274435584000000000000000000, 18011357710498321908881994832212360081640749122627567616000000000000000000000000

Offset: 0

Matthew Campbell, Jul 30 2015

			a(3) = (Hyperfactorial(3)/Superfactorial(3)) * (Hyperfactorial(2)/Superfactorial(2)) * (Hyperfactorial(1)/Superfactorial(1)) * (Hyperfactorial(0)/Superfactorial(0)) = ((3^3 * 2^2 * 1^1)/(3! * 2! * 1!)) * ((2^2 * 1^1)/(2!*1!)) * (1^1/1!) * 1 = ((27 * 4)/(6 * 2)) * (4/2) * 1 = (108/12) * (4/2) = 9 * 2 = 18.

Matthew Campbell, Table of n, a(n) for n = 0..24

Cf. A000178, A001142, A002109, A055462, A125760, A255269.

Table[Product[Hyperfactorial[n]/BarnesG[n+2], {n, 0, m}], {m, 0, 12}]
Table[BarnesG[n+2]^(n-1) / Product[BarnesG[k]^3, {k, 1, n + 1}], {n, 0, 12}] (* Vaclav Kotesovec, Nov 19 2023 *)

a001142(n) = prod(m=1, n, binomial(n, m));
a(n) = prod(k=0, n, a001142(k)); \\ Michel Marcus, Aug 06 2015

a(n) = A125760(n)/A055462(n).
a(n) = Product_{k=0..n} A001142(k).
a(n) = Product_{k=0..n} hyperfactorial(k)/superfactorial(k).
a(n) = Product_{i=1..n} (Product_{j=1..i} binomial(i,j)). - Pedro Caceres, Apr 13 2019
From Vaclav Kotesovec, Nov 19 2023: (Start)
a(n) = BarnesG(n+2)^(n-1) / Product_{k=1..n+1} BarnesG(k)^3.
a(n) ~ A^(2*n + 5/2) * exp(n^3/6 + 7*n^2/8 + 5*n/6 - 3*zeta(3)/(8*Pi^2) - 1/8) / ((2*Pi)^(n^2/4 + 3*n/4 + 1/2) * n^(n^2/4 + 7*n/12 + 7/24)), where A is the Glaisher-Kinkelin constant A074962. (End)

A304902 Let (P,<) be the strict partial order on the subsets of {1,2,...,n} ordered by their cardinality. Then a(n) is the number of paths of any length from {} to {1,2,...,n}.

Original entry on oeis.org

1, 1, 3, 16, 175, 4356, 263424, 40144896, 15714084159, 15953234222500, 42223789335548788, 292262228709213966336, 5302397936652484482131200, 252622720869371754406993137664, 31660291085217875120800516475520000, 10454334647424614439930776175842716286976

Offset: 0

Geoffrey Critzer, May 20 2018

nonn

A001142 counts such paths of length n.

A000670 counts such paths under the inclusion relation.

Alois P. Heinz, Table of n, a(n) for n = 0..69

Cf. A000670, A001142.

b:= proc(n, k) option remember; `if`(k=0, 1,
      add(b(n, j), j=0..k-1)*binomial(n, k))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);  # Alois P. Heinz, May 20 2018

Table[f[list_] := Apply[Times, Map[Binomial[n, #] &, list]];
Total[Map[f, Map[Accumulate,Level[Map[Permutations, Partitions[n]], {2}]]]], {n, 0, 15}]

A334038 a(n) = Product_{p<=n, p prime} binomial(n,p).

Original entry on oeis.org

1, 1, 1, 3, 24, 100, 1800, 15435, 702464, 13716864, 163296000, 1383574500, 109294479360, 3842829083808, 1159801183597056, 132320316074821875, 8213884352593920000, 327816138093181337600, 167079259535068179726336, 34044607357920579594754944

Offset: 0

Om R. Patel, Apr 13 2020

nonn

			For n=2, p=2: a(n) = C(2,2) = 1.
For n=3, p=2,3: a(n) = C(3,2) * C(3,3) = 3.
For n=4, p=2,3: a(n) = C(4,2) * C(4,3) = 24.

Alois P. Heinz, Table of n, a(n) for n = 0..145

Cf. A000040, A001142, A010051, A121497.

a:= n-> mul(`if`(isprime(p), binomial(n, p), 1), p=2..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 13 2020

a(n) = prod(k=1, n, if (isprime(k), binomial(n, k), 1)); \\ Michel Marcus, Apr 13 2020

a(n)=my(s); forprime(p=2,n, s*=binomial(n,p)); s \\ Charles R Greathouse IV, Apr 13 2020

A191510 Product of terms in n-th row of A132818.

Original entry on oeis.org

1, 9, 648, 360000, 1518750000, 48243443062500, 11480517255997440000, 20400479323264014247526400, 270090559531318533654528000000000, 26599911685677709861296622500000000000000, 19464564507161243794359748945629699456000000000000

Offset: 1

Harlan J. Brothers, Jun 04 2011

Lim_{n -> inf} (a(n)*a(n+2))/a(n+1)^2 = e^2. Like A168510, this limit is asymptotic from above.

			For n=3, row 3 of A132818 = {6,18,6} and a(3)=648.

H. J. Brothers and J. A. Knox, New closed-form approximations to the logarithmic constant e, Math. Intelligencer, Vol. 20, No. 4, (1998), 25-29.

Cf. A132818, A002457. Related to e as in the cases of A168510 and A001142.

Table[Product[Product[((k + 1)/(k - 1))^k, {k, 2, j}], {j, 1, n}], {n, 1, 11}]
Table[(n + 1)^n * Hyperfactorial[n]^2 / (2^n * BarnesG[n+2]^2), {n, 1, 12}] (* Vaclav Kotesovec, Jul 11 2015 *)

a(n)=product[product[((k + 1)/(k - 1))^k, {k, 2, j}], {j, 1, n}].

a(n) ~ A^4 * exp(n^2 + 2*n + 5/6) / (n^(2/3) * 2^(2*n+1) * Pi^(n+1)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 11 2015

A265944 Absolute value of the determinant of the matrix whose terms are fibonacci(m+r+s)^(n) with 0 <= r, s <=n, for any m.

Original entry on oeis.org

1, 2, 36, 13824, 324000000, 1209323520000000, 1923567501916569600000000, 3436011282355888738787131392000000000, 18204541483393435808637499286914987185930240000000000, 753091424970084722185225494963366011108371967508480000000000000000000000

Offset: 1

Michel Marcus, Dec 23 2015

nonn

L. Carlitz, Some Determinants Containing Powers of Fibonacci Numbers, The Fibonacci Quarterly, 4.2 (1966), 129-134.
Eric Rowland and Jesus Sistos Barron, Complexity of powers of a constant-recursive sequence, arXiv:2501.14643 [math.NT], 2025. See p. 6.
Aram Tangboonduangjit and Thotsaporn Thanatipanonda, Determinants Containing Powers of Generalized Fibonacci Numbers, arXiv:1512.07025 [math.CO], 2015.

Cf. A001142, A152686.

a(n) = prod(j=0, n, binomial(n, j)) * prod(j=1,n, fibonacci(j)^(n-j+1))^2;

a(n) = (Product_{j=0..n} binomial(n,j)) * (Product_{j=1..n} fibonacci(j)^(n-j+1))^2.

a(n) = A001142(n)*A152686(n)^2.

A355635 Triangle read by rows. Row n gives the coefficients of Product_{k=0..n-1} (x - binomial(n-1,k)) expanded in decreasing powers of x, with row 0 = {1}.

Original entry on oeis.org

1, 1, -1, 1, -2, 1, 1, -4, 5, -2, 1, -8, 22, -24, 9, 1, -16, 93, -238, 256, -96, 1, -32, 386, -2180, 5825, -6500, 2500, 1, -64, 1586, -19184, 117561, -345600, 407700, -162000, 1, -128, 6476, -164864, 2229206, -15585920, 51583084, -64538880, 26471025

Offset: 0

Thomas Scheuerle, Jul 11 2022

Without signs the triangle of elementary symmetric functions of the terms binomial(n,j), j=0..n.

			The triangle begins:
  1;
  1,  -1;
  1,  -2,   1;
  1,  -4,   5,    -2;
  1,  -8,  22,   -24,    9;
  1, -16,  93,  -238,  256,   -96;
  1, -32, 386, -2180, 5825, -6500, 2500;
  ...
Row 4: x^4 - 8*x^3 + 22*x^2 - 24*x + 9 = (x-1)*(x-4)*(x-6)*(x-4)*(x-1).

Cf. A000079, A000346, A025131, A025133, A025134, A025135.

Cf. A001142 (right diagonal unsigned).

T(n, k) = polcoeff(prod(m=0, n, (x-binomial(n-1, m))), n-k+1);

T(n, 0) = 1.
T(n, 1) = -2^(n-1), for n > 0.
T(n, 2) = A000346(n-2), for n > 1.
T(n, 3) = -A025131(n-1), for n > 1.
T(n, 4) = A025133(n-1), for n > 1.
T(n, n) = (-1)^n*A001142(n-1), for n > 0.
T(n+1, n) = (-1)^n*A025134(n).
T(n+2, n) = (-1)^n*A025135(n).

cp's OEIS Frontend

A132858 Composite "antimutinous" numbers. An antimutinous number is an integer m > 1 where m/p^k < p, where p is the largest prime divisor of m and p^k is the largest power of p dividing m.

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A260610 Suprafactorials: Product of first n hyperfactorials divided by the product of the first n superfactorials.

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A304902 Let (P,<) be the strict partial order on the subsets of {1,2,...,n} ordered by their cardinality. Then a(n) is the number of paths of any length from {} to {1,2,...,n}.

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A334038 a(n) = Product_{p<=n, p prime} binomial(n,p).

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A191510 Product of terms in n-th row of A132818.

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A265944 Absolute value of the determinant of the matrix whose terms are fibonacci(m+r+s)^(n) with 0 <= r, s <=n, for any m.

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A355635 Triangle read by rows. Row n gives the coefficients of Product_{k=0..n-1} (x - binomial(n-1,k)) expanded in decreasing powers of x, with row 0 = {1}.

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