A003046 -id:A003046 - cp's OEIS frontend

A294373 Product of first n Bell numbers.

1, 1, 2, 10, 150, 7800, 1583400, 1388641800, 5748977052000, 121573617718644000, 14099500314919737900000, 9567497928695086546803000000, 40313580569855830588349480391000000, 1114446238307803607782300144651734867000000

Offset: 0

Vaclav Kotesovec, Oct 29 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..49

Cf. A000110, A003266, A070825, A003046, A058807, A058808, A342166, A342170.

B:= map(combinat:-bell, [$0..19]):
map(i -> convert(B[1..i],`*`),[$1..20]); # Robert Israel, Oct 29 2017

Table[Product[BellB[k], {k, 0, n}], {n, 0, 15}]

log(a(n)) ~ n^2 * LambertW(n)/2 * (1 - 3/(2*LambertW(n)) + 3/(2*LambertW(n)^2) + 1/(4*LambertW(n)^3)). - Vaclav Kotesovec, Feb 26 2021

A007686 Prime(n)...a(n) is the least product of consecutive primes which is non-deficient.

Original entry on oeis.org

3, 13, 31, 73, 149, 233, 367, 521, 733, 991, 1249, 1579, 1949, 2341, 2791, 3343, 3881, 4481, 5147, 5849, 6619, 7499, 8387, 9341, 10321, 11411, 12517, 13709, 15013, 16363, 17881, 19381, 20873, 22369, 24007, 25763, 27611, 29399, 31357

Offset: 1

Walter Nissen

nonn

Differs from A007708 only for n=1. - Michel Marcus, Mar 10 2013

a(n) is approximately n^2 log^2 n. - Charles R Greathouse IV, Feb 26 2025

Amiram Eldar, Table of n, a(n) for n = 1..1000
H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy) [An annotation on p. 81 references this A-number, but the sequence with that annotation is actually A003046 (the last term is erroneous), unrelated to this entry. - _Andrey Zabolotskiy_, Feb 26 2025]

Cf. A005100, A007684, A007702, A007708.

a[n_] := Module[{p = Prime[n]}, r = 1 + 1/p; While[r < 2,  p = NextPrime[p]; r *= 1 + 1/p]; p]; Array[a, 39] (* Amiram Eldar, Jun 29 2019 *)

a(n) = {p = prime(n); sig = p+1; prd = p; while (sig < 2*prd, p = nextprime(p+1); sig *= p+1; prd *= p;); return (p);} \\ Michel Marcus, Mar 10 2013

More terms from Don Reble, Nov 10 2005

A342170 Product of first n little Schröder numbers.

Original entry on oeis.org

1, 1, 3, 33, 1485, 292545, 264168135, 1130375449665, 23503896724884345, 2422053053602606867905, 1256704025339194996874320395, 3326147448057830199712191898815585, 45398150793225628820115544929795174823365, 3225056167710201318911738099365978237877235350145

Offset: 0

Vaclav Kotesovec, Mar 03 2021

nonn

Eric Weisstein's World of Mathematics, Super Catalan Number.
Wikipedia, Schroeder-Hipparchus numbers.

Cf. A001003, A003046, A003266, A070825, A294373, A342166.

b:= proc(n) option remember; `if`(n<2, 1,
      ((6*n-3)*b(n-1)-(n-2)*b(n-2))/(n+1))
    end:
a:= proc(n) a(n):=`if`(n=0, 1, a(n-1)*b(n)) end:
seq(a(n), n=0..15);  # Alois P. Heinz, Mar 03 2021

Table[Product[Hypergeometric2F1[1-k, k+2, 2, -1], {k, 1, n}], {n, 0, 15}]
FoldList[Times, 1, Table[Hypergeometric2F1[1 - n, n + 2, 2, -1], {n, 1, 15}]]

a(n) = Product_{k=1..n} A001003(k).

a(n) ~ c * (1 + sqrt(2))^(n*(n+2)) * exp(3*n/2) / (2^((7*n + 3)/4) * Pi^((2*n + 3)/4) * n^(3*n/2 + 3/2 + 9/(16*sqrt(2)))), where c = 0.89405100528141459535141257102427907468205556782800836208733677564241771912...

A069640 Let M_n be the n X n matrix with M_n(i,j)=1/(i+j+1); then a(n)=1/det(M_n).

Original entry on oeis.org

3, 240, 378000, 10668672000, 5175372787200000, 42202225467872870400000, 5708700736339601341845504000000, 12701009683686045652926579789004800000000, 462068939479146913162956288390362787269836800000000

Offset: 1

Benoit Cloitre, Apr 21 2002

Wolfram Research, 1991 Mathematica Conference, Elementary Tutorial Notes, Section 1, Introduction to Mathematica, Paul Abbott, page 19.

Robert Israel, Table of n, a(n) for n = 1..40

Cf. A000108, A003046, A005249, A067689, A074962.

seq(1/LinearAlgebra:-Determinant(LinearAlgebra:-HilbertMatrix(n,n,-1)),n=1..10); # Robert Israel, Sep 26 2018

Hilbert[n_Integer] := Table[1/(i + j + 1), {i, n}, {j, n}]; Table[ 1 / Det[ Hilbert[n]], {n, 1, 8}] (* Robert G. Wilson v, Mar 13 2004 *)
Table[(2*n+1)!!*(n!*Product[(2*k)!/k!/(k+1)!,{k,0,n}])^2,{n,1,11}] (* Alexander Adamchuk, May 17 2006 *)
Table[2^(2*n^2+2*n-1/12) * Glaisher^3 * BarnesG[n+3/2]^2 *(n!)^2 *(2*n+1)!!/(E^(1/4)*Pi^(n+1/2)*BarnesG[n+3]^2), {n, 1, 11}] (* Vaclav Kotesovec, Mar 09 2014 *)

for(n=1,10,print1(1/matdet(matrix(n,n,i,j,1/(i+j+1))),","))

a(n) = (2*n+1)!!*(n!*Product[(2*k)!/k!/(k+1)!,{k,0,n}])^2. a(n) = (2*n+1)!!*(n!*A003046(n))^2, where A003046(n)is the Product of first n Catalan numbers A000108(n). a(n) = (2*n+1)!*n!/(2^n)*A003046(n)^2. - Alexander Adamchuk, May 17 2006

a(n) ~ A^3 * 2^(2*n^2+3*n+11/12) / (exp(1/4) * n^(7/4) * Pi^(n+1)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 13 2014

A268646 O.g.f.: 1/(1 - C(1)x/(1 - C(2)x/(1 - C(3)x/(1 - C(4)x/(1 - C(5)x/(1 - C(6)x/(1 -...))))))), a continued fraction, where C(n) are the Catalan numbers A000108.

Original entry on oeis.org

1, 1, 3, 19, 277, 11081, 1383243, 569441699, 791393701997, 3770885471695081, 62402464265309818563, 3626978195203590614565619, 747715555141652980441024051237, 551447343768097359581617325419468841, 1465935896222119146302554598601016693710363

Offset: 0

Benedict W. J. Irwin, Feb 09 2016

Seiichi Manyama, Table of n, a(n) for n = 0..61

Cf. A000108, A003046.

Table[SeriesCoefficient[Series[1/(1+ContinuedFractionK[-CatalanNumber[k]*x,1, {k,1,50}]),{x,0,50}],n],{n,1,50}]

G.f.: 1/(1 - x/(1 - 2x/(1 - 5x/(1 - 14x/(1 - 42x/(1 -...)))))), by definition.

a(n) ~ c * A003046(n) ~ c * A^(3/2) * 2^(n^2+n-19/24) * exp(3*n/2-1/8) / (n^(3*n/2+15/8) * Pi^(n/2+1)), where A is the Glaisher-Kinkelin constant A074962 and c = 1/Product_{k>=1} (1 - 1/4^k) = 1/QPochhammer[1/4] = 1.452353642449597... - Vaclav Kotesovec, Aug 26 2017

a(0) = 1 added by Peter Bala, Apr 17 2017

A342166 Product of first n Fubini numbers.

Original entry on oeis.org

1, 1, 3, 39, 2925, 1582425, 7410496275, 350464600333575, 191295845123076910125, 1355763582602823185129417625, 138623522325287867599380791765497875, 224935042709004795568466587349227029537282375, 6318777956744220129890735589019782971247629409914638125

Offset: 0

Vaclav Kotesovec, Mar 03 2021

nonn

Wikipedia, Ordered Bell number

Cf. A000670, A003046, A003266, A070825, A294373, A342170.

g:= proc(n) option remember; `if`(n=0, 1,
      add(g(n-j)*binomial(n, j), j=1..n))
    end:
a:= proc(n) option remember; `if`(n=0, 1, a(n-1)*g(n)) end:
seq(a(n), n=0..15);  # Alois P. Heinz, Mar 03 2021

Table[Product[Sum[j!*StirlingS2[k, j], {j, 0, k}], {k, 1, n}], {n, 0, 12}]
Table[Product[PolyLog[-k, 1/2]/2, {k, 1, n}], {n, 0, 12}]
FoldList[Times, 1, Table[PolyLog[-n, 1/2]/2, {n, 1, 12}]]

a(n) = Product_{k=1..n} A000670(k).
a(n) ~ c * BarnesG(n+2) / (2^n * log(2)^(n*(n+3)/2)), where c = 0.960303470666951851619546415046950178638511457142008903473074598398282549...
a(n) ~ c * Pi^((n+1)/2) * n^(n^2/2 + n + 5/12) / (A * 2^((n-1)/2) * exp(3*n^2/4 + n - 1/12) * log(2)^(n*(n+3)/2)), where A is the Glaisher-Kinkelin constant A074962.

A386649 Product of first n central trinomial coefficients (A002426) for n > 0 with a(0) = 1.

Original entry on oeis.org

1, 1, 3, 21, 399, 20349, 2869209, 1127599137, 1248252244659, 3918263795984601, 35080215765450132753, 899912775031092255512709, 66403663756769266442027284401, 14140062564030204365431731967633341, 8713488333644640745496899895218790824407

Offset: 0

Paul D. Hanna, Aug 08 2025

nonn

Conjecture: a(n) = A214589(n) - 2 for n >= 1, where A214589(n) is the number of n X n X n triangular 0..2 arrays with every horizontal row having the same average value.

			The central trinomial coefficients A002426(n) = [x^n] (1 + x + x^2)^n for n >= 0 begin [1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, ...], where a(n) equals the product of the terms A002426(0) through A002426(n).

Paul D. Hanna, Table of n, a(n) for n = 0..70

Cf. A214589, A386650, A342177, A003046, A002426.

Table[Product[3^k * Hypergeometric2F1[1/2, -k, 1, 4/3], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Aug 09 2025 *)

{a(n) = prod(k=0,n, polcoef((1 + x + x^2)^k, k) )}
for(n=0,15,print1(a(n),", "))

a(n) = Product_{k=0..n} A002426(k) for n >= 0.

a(n) ~ c * 3^((n-1)*(n+3)/2) * exp(n/2) / (2^(n - 3/4) * Pi^(n/2 - 1/4) * n^(n/2 + 7/16)), where c = 1.123782729130753266489882099159237662230713685736... - Vaclav Kotesovec, Aug 09 2025

A386650 Product of first n quadrinomial coefficients (A005725) for n > 0 with a(0) = 1.

Original entry on oeis.org

1, 1, 3, 30, 930, 93930, 31560480, 35600221440, 136099646565120, 1776236487321381120, 79580723341459838319360, 12296654209275691297430868480, 6578267322410960919238807125534720, 12223446894741861497849104893155093176320, 79112201841847644246811045518121813092796661760

Offset: 0

Paul D. Hanna, Aug 08 2025

nonn

Conjecture: 2*a(n) = A214590(n) - 2 for n >= 1, where A214590(n) is the number of nXnXn triangular 0..3 arrays with every horizontal row having the same average value.

			The quadrinomial coefficients A005725(n) = [x^n] (1 + x + x^2 + x^3)^n for n >= 0 begin [1, 1, 3, 10, 31, 101, 336, 1128, 3823, ...], where a(n) equals the product of the terms A005725(0) through A005725(n).

Paul D. Hanna, Table of n, a(n) for n = 0..70

Cf. A214590, A386649, A342177, A003046, A005725.

Table[Product[HypergeometricPFQ[{(1-k)/2, -k, -k/2}, {1/2, 1}, -1], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Aug 09 2025 *)

{a(n) = prod(k=0, n, polcoef((1 + x + x^2 + x^3)^k, k) )}
for(n=0, 15, print1(a(n), ", "))

a(n) = Product_{k=0..n} A005725(k) for n >= 0.

a(n) ~ c * exp(n/2) * (11 + 217/(6371 + 624*sqrt(78))^(1/3) + (6371 + 624*sqrt(78))^(1/3))^(-1 + n/2 + n^2/2) * ((39 + (4563 - 78*sqrt(78))^(1/3) + (4563 + 78*sqrt(78))^(1/3))/13)^(n/2) / (2^(-11/4 + 2*n + n^2) * 3^((-3 + 2*n + n^2)/2) * Pi^(n/2 + 1/4) * n^((4290 - 1421*78^(2/3)/(804726 - 73709*sqrt(78))^(1/3) - (78*(804726 - 73709*sqrt(78)))^(1/3) + 4056*n)/8112)), where c = 0.77060824350557924602665408964165291884080801923663... - Vaclav Kotesovec, Aug 09 2025

A003047 a(n) = Catalan(n) * Product_{k = 0..n-1} a(k).

Original entry on oeis.org

1, 1, 2, 10, 280, 235200, 173859840000, 98238542885683200000000, 32169371027674057560745102540800000000000000000, 3518552669874927170883258602839130084576857453953842493259776000000000000000000000000000000000

Offset: 1

N. J. A. Sloane

D.-Z. Du and F. K. Hwang, Combinatorial Group Testing and Its Applications, World Scientific, 2nd ed., 2000; p. 36.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. W. Moon, R. K. Guy, and N. J. A. Sloane, Correspondence, 1973.
J. W. Moon and M. Sobel, Enumerating a class of nested group testing procedures, J. Combin. Theory, B23 (1977), 184-188.

See A028580 for another version. Cf. A000108, A003046.

a(n) = ((4*n-6)*a(n-1)^2)/n, for n > 1. - Christian Krause, Oct 13 2023

A051575 a(n) = LCM { Catalan(0), ..., Catalan(n) }.

Original entry on oeis.org

1, 1, 2, 10, 70, 210, 4620, 60060, 60060, 1021020, 19399380, 19399380, 446185740, 2230928700, 13385572200, 388181593800, 12033629407800, 12033629407800, 12033629407800, 445244288088600, 445244288088600, 18255015811632600

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A000108, A003046, A078491.

LCM@@@CatalanNumber@Range[0, Range[0, 20]] (* Vladimir Reshetnikov, Nov 17 2015 *)

a(n) = if(n>1, lcm(a(n-1), binomial(2*n, n)/(n+1)), 1) \\ David A. Corneth, Aug 23 2016

cp's OEIS Frontend

A294373 Product of first n Bell numbers.

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A007686 Prime(n)*...*a(n) is the least product of consecutive primes which is non-deficient.

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A342170 Product of first n little Schröder numbers.

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A069640 Let M_n be the n X n matrix with M_n(i,j)=1/(i+j+1); then a(n)=1/det(M_n).

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A268646 O.g.f.: 1/(1 - C(1)x/(1 - C(2)x/(1 - C(3)x/(1 - C(4)x/(1 - C(5)x/(1 - C(6)x/(1 -...))))))), a continued fraction, where C(n) are the Catalan numbers A000108.

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Extensions

A342166 Product of first n Fubini numbers.

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A386649 Product of first n central trinomial coefficients (A002426) for n > 0 with a(0) = 1.

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A386650 Product of first n quadrinomial coefficients (A005725) for n > 0 with a(0) = 1.

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A007686 Prime(n)...a(n) is the least product of consecutive primes which is non-deficient.