A007685 -id:A007685 - cp's OEIS frontend

A367568 a(n) = Product_{k=0..n} (4*k)! / k!^4.

1, 24, 60480, 22353408000, 1409672968704000000, 16539333509029163728896000000, 38185078618454141182825889242546176000000, 18043150250179542387558306410182977707728856678400000000, 1796395750154420920494206475343190362781863323574704301041254400000000000

Offset: 0

Vaclav Kotesovec, Nov 23 2023

nonn

Cf. A000178, A008977, A268505, A168488, A268196, A262261.

Cf. A007685, A367567, A367569, A367570, A367571.

Table[Product[(4*k)!/k!^4, {k, 0, n}], {n, 0, 10}]
Table[Product[Binomial[4*k,k] * Binomial[3*k,k] * Binomial[2*k,k], {k, 0, n}], {n, 0, 10}]

a(n) = Product_{k=0..n} binomial(4*k,k) * binomial(3*k,k) * binomial(2*k,k).
a(n) = A268505(n) / A000178(n)^4.
a(n) = A268505(n) / A168488(n).
a(n) = A007685(n) * A268196(n) * A262261(n).
a(n) ~ A^(15/4) * sqrt(Gamma(1/4)) * 2^(4*n^2 + 7*n/2 - 7/6) * exp(3*n/2 - 5/16) / (n^(3*n/2 + 17/16) * Pi^(3*n/2 + 7/4)), where A is the Glaisher-Kinkelin constant A074962.

A367569 a(n) = Product_{k=0..n} (5*k)! / k!^5.

Original entry on oeis.org

1, 120, 13608000, 2288430144000000, 699207483978843840000000000, 435858496811697532778806061260800000000000, 597507154003470929939550139366865942134606725120000000000000, 1898554530971015145216561379837863419725314413457243266261094236160000000000000000

Offset: 0

Vaclav Kotesovec, Nov 23 2023

nonn

Cf. A000178, A008978, A268506.

Cf. A007685, A367567, A367568, A367570, A367571.

Table[Product[(5*k)!/k!^5, {k, 0, n}], {n, 0, 10}]
Table[Product[Binomial[5*k,k] * Binomial[4*k,k] * Binomial[3*k,k] * Binomial[2*k,k], {k, 0, n}], {n, 0, 10}]

a(n) = Product_{k=0..n} binomial(5*k,k) * binomial(4*k,k) * binomial(3*k,k) * binomial(2*k,k).
a(n) = A268506(n) / A000178(n)^5.
a(n) ~ A^(24/5) * Gamma(1/5)^(3/5) * Gamma(2/5)^(2/5) * Gamma(3/5)^(1/5) * 5^(5*n^2/2 + 3*n + 23/60) * exp(2*n - 2/5) / (n^(2*n + 7/5) * (2*Pi)^(2*n + 13/5)), where A is the Glaisher-Kinkelin constant A074962.
Equivalently, a(n) ~ A^(24/5) * Gamma(1/5)^(3/5) * Gamma(2/5)^(1/5) * 5^(5*n^2/2 + 3*n + 1/3) * exp(2*n - 2/5) / ((1 + sqrt(5))^(1/10) * 2^(2*n + 23/10) * Pi^(2*n + 12/5) * n^(2*n + 7/5)).

A367570 a(n) = Product_{k=0..n} (6*k)! / k!^6.

Original entry on oeis.org

1, 720, 5388768000, 739474163011584000000, 2400828978003787120431882240000000000, 213271990853093812884314351984207293234859212800000000000, 569474121824212834327144127568532894901251393782268174537457286512640000000000000

Offset: 0

Vaclav Kotesovec, Nov 23 2023

nonn

Cf. A000178, A008979, A271946.

Cf. A007685, A367567, A367568, A367569, A367571.

Table[Product[(6*k)!/k!^6, {k, 0, n}], {n, 0, 10}]
Table[Product[Binomial[6*k,k] * Binomial[5*k,k] * Binomial[4*k,k] * Binomial[3*k,k] * Binomial[2*k,k], {k, 0, n}], {n, 0, 10}]

a(n) = Product_{k=0..n} binomial(6*k,k) * binomial(5*k,k) * binomial(4*k,k) * binomial(3*k,k) * binomial(2*k,k).
a(n) = A271946(n) / A000178(n)^6.
a(n) ~ A^(35/6) * Gamma(1/3)^(5/3) * 2^(3*n^2 + n - 215/72) * 3^(3*n^2 + 7*n/2 + 47/72) * exp(5*n/2 - 35/72) / (n^(5*n/2 + 125/72) * Pi^(5*n/2 + 10/3)), where A is the Glaisher-Kinkelin constant A074962.

A367571 a(n) = Product_{k=0..n} (7*k)! / k!^7.

Original entry on oeis.org

1, 5040, 3432645216000, 626489905645044080640000000, 41646279370357699257014919153469440000000000000, 1200992054275801322636044235924808416678612164215512865177600000000000000

Offset: 0

Vaclav Kotesovec, Nov 23 2023

nonn

In general, for m > 1, Product_{k=0..n} (m*k)! / k!^m ~ A^(m - 1/m) * exp(m*n/2 - m/12 + 1/(12*m) - n/2) * m^(m*n^2/2 + m*n/2 - 1/(12*m) + n/2) * n^(-m*n/2 - m/3 + 1/(12*m) + n/2 + 1/4) * (2*Pi)^(-m*n/2 - m/4 + n/2 + 1/4) / Product_{j=1..m-1} Gamma(j/m)^(j/m), where A is the Glaisher-Kinkelin constant A074962.

Cf. A000178, A071549, A271947.

Cf. A007685, A367567, A367568, A367569, A367570.

Table[Product[(7*k)!/k!^7, {k, 0, n}], {n, 0, 10}]
Table[Product[Binomial[7*k,k] * Binomial[6*k,k] * Binomial[5*k,k] * Binomial[4*k,k] * Binomial[3*k,k] * Binomial[2*k,k], {k, 0, n}], {n, 0, 10}]

a(n) = Product_{k=0..n} binomial(7*k,k) * binomial(6*k,k) * binomial(5*k,k) * binomial(4*k,k) * binomial(3*k,k) * binomial(2*k,k).
a(n) = A271947(n) / A000178(n)^7.
a(n) ~ A^(48/7) * 7^(7*n^2/2 + 4*n - 1/84) * exp(3*n - 4/7) / (Gamma(1/7)^(1/7) * Gamma(2/7)^(2/7) * Gamma(3/7)^(3/7) * Gamma(4/7)^(4/7) * Gamma(5/7)^(5/7) * Gamma(6/7)^(6/7) * n^(3*n + 29/14) * (2*Pi)^(3*n + 3/2)), where A is the Glaisher-Kinkelin constant A074962.

A324566 a(n) = Product_{i=0..n, j=0..n} (binomial(2i, i) + binomial(2j, j)).

Original entry on oeis.org

2, 72, 2709504, 15637972132823040, 2676716427588188879089135779840000, 2844659827809077182003221596452154340416256794895644098560000

Offset: 0

Vaclav Kotesovec, Mar 07 2019

nonn

Cf. A000984, A007685, A324565, A324567.

Table[Product[Binomial[2*i, i] + Binomial[2*j, j], {i, 0, n}, {j, 0, n}], {n, 0, 7}]

a(n) ~ c * d^n * 2^(4*n^3/3 + 3*n^2) * exp(n^2/4 + n) / (Pi^(n^2/2) * n^(n^2/2 + n + alfa), where
d = 1.7552437670085151253596298317411616254914764275962173772854884197280838467...
alfa = 0.0720289309157636194865104906223497131817905230497828189113545663815669...
c = 1.3392511999007422713775779654793976642559410116533455108049762435308676715...

A338550 Number of binary trees of height n such that the number of nodes at depth d equals d+1 for every d = 0..n.

Original entry on oeis.org

1, 1, 4, 60, 3360, 705600, 558835200, 1678182105600, 19198403288064000, 840083731079104512000, 141100463472046393835520000, 91242050302344912388163665920000, 227753296409896438988240405704212480000, 2199573010737856838816729366169572868096000000, 82356764599728553816070191604819734458909327360000000

Offset: 0

Marcel K. Goh, Nov 02 2020

To satisfy the constraint, there must be n+1 nodes at depth n, and there are 2n allowed slots for a new node.

A binary tree with such a level profile contains A000217(n+1) nodes.

Cf. A000217, A007685, A203471, A296589.

Table[Product[Binomial[2k,k+1],{k,n}],{n,0,14}] (* or *)
Table[2^(n^2+n-1/24)Glaisher^(3/2)Pi^(-1/4-n/2)BarnesG[3/2+n]Gamma[1+n]/(Exp[1/8]BarnesG[3+n]),{n,0,14}] (* Stefano Spezia, Nov 02 2020 *)

a(n) = binomial(2*n,n+1)*a(n-1), a(0)=1.
a(n) = Product_{k=1..n} binomial(2*k,k+1).
a(n) = 2^(n^2+n-1/24)*A^(3/2)*Pi^(-1/4-n/2)*G(3/2 + n)*Gamma(1 + n)/(exp(1/8)*G(3 + n)) where A is the Glaisher-Kinkelin constant and G is the Barnes G function. - Stefano Spezia, Nov 02 2020
a(n) ~ A^(3/2) * 2^(-7/24 + n + n^2) * exp(-1/8 + n/2) / (n^(11/8 + n/2) * Pi^((n+1)/2)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 29 2023
a(n) = Product_{1 <= j <= i <= n-1}  (i + j + 2)/(i - j + 1). - Peter Bala, Oct 25 2024

A324567 a(n) = Product_{i=0..n, j=0..n} (binomial(2i, j) + binomial(2j, i)).

Original entry on oeis.org

2, 8, 2400, 1247616000, 47391629172572160000, 5433273487668074503912921497600000000, 84476763043100284572577776893541858819157327099409203200000000

Offset: 0

Vaclav Kotesovec, Mar 07 2019

nonn

Cf. A000984, A007685, A324566, A324568.

Table[Product[Binomial[2*i, j] + Binomial[2*j, i], {i, 0, n}, {j, 0, n}], {n, 0, 7}]

a(n) ~ c * d^n * exp(2*n^3/3 + 9*n^2/4) / (2^(n^2) * Pi^(n^2/2) * n^(n^2/2 + 5*n/6 - alfa)), where
d = 2.631186542501652610455988727776308850706336468478433795517960445788077147...
alfa = 0.094637735750903047180156176044316953484876143616883847755312100285099...
c = 0.121223472988717836553569221604723373713338356783096...

A324568 a(n) = Sum_{i=0..n, j=0..n} (binomial(2i, j) + binomial(2j, i)).

Original entry on oeis.org

2, 8, 32, 124, 482, 1882, 7380, 29036, 114530, 452638, 1791638, 7100430, 28167986, 111837902, 444351292, 1766536044, 7026526226, 27960911422, 111308958942, 443258277254, 1765690504666, 7035402933402, 28039342445582, 111773962249054, 445654589001882

Offset: 0

Vaclav Kotesovec, Mar 07 2019

nonn

Robert Israel, Table of n, a(n) for n = 0..1653

Cf. A000984, A006134, A007685, A324566, A324567.

R:= 2: r:= 2;
for n from 1 to 30 do
  v:= 2*binomial(2*n,n) + 2*add(binomial(2*n,j),j=0..n-1) + 2*add(binomial(2*j,n),j=ceil(n/2) .. n-1);
  r:= r+v;
  R:= R,r;
od:
R; # Robert Israel, Mar 02 2025

Table[Sum[Binomial[2*i, j] + Binomial[2*j, i], {i, 0, n}, {j, 0, n}], {n, 0, 30}]

Recurrence: 2*(n+1)*(5*n^2 - 21*n + 20)*a(n) = (85*n^3 - 332*n^2 + 243*n + 100)*a(n-1) - 3*(65*n^3 - 298*n^2 + 377*n - 100)*a(n-2) + 2*(20*n^3 - 109*n^2 + 191*n - 100)*a(n-3) + 8*(2*n - 5)*(5*n^2 - 11*n + 4)*a(n-4).

a(n) ~ 4^(n+1)/3 * (1 + 5/(3*sqrt(Pi*n))).

A371646 a(n) = Product_{k=0..n} binomial(n^3, k^3).

Original entry on oeis.org

1, 1, 8, 59942025, 239830737497318918172122578944, 788243862228623056807478850630904903414781894638966172447366478063616699218750

Offset: 0

Vaclav Kotesovec, Mar 31 2024

nonn

Cf. A001142, A371603.

Cf. A007685, A255358, A371468.

Table[Product[Binomial[n^3, k^3], {k, 0, n}], {n, 0, 6}]

a(n) ~ c * exp((9/4 - sqrt(3)*Pi/8)*n^4 + (3*zeta(3)/(4*Pi^2) - Pi/(4*sqrt(3)) + 3)*n) / ((2*Pi)^(n/2) * A^(3*n^2) * 3^(9*n^4/8 - n^2/4 + 3*n/4) * n^(n^2/4 + 3*n/2 - 8/15)), where c = 0.498332919... and A is the Glaisher-Kinkelin constant A074962.

A374891 Obverse convolution (1)**A000984; see Comments.

Original entry on oeis.org

2, 6, 42, 882, 62622, 15843366, 14655113550, 50311004817150, 647552943001537650, 31484671641677762080650, 5817013478501458288734652050, 4103513269179719224996951799587650, 11096544131445222000310082187517540861050

Offset: 0

Clark Kimberling, Jul 31 2024

nonn

See A374848 for the definition of obverse convolution and a guide to related sequences. This is a divisibility sequence (see Formula).

Cf. A000012, A000984, A007685, A374848.

s[n_] := 1; t[n_] := Binomial[2 n, n];
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[u[n], {n, 0, 20}]

a(n+1) = a(n)*A244174(n+1) for n>=0 (conjectured) = a(n)*A323230(n+2) for n>=0 (conjectured).

a(n) ~ c * A007685(n), where c = Product_{k=0..oo} (1 + 1/binomial(2*k,k)) = 3.74782908533723753117687910314018231428739915473496578523053032212205053... - Vaclav Kotesovec, Jul 31 2024

cp's OEIS Frontend

A367568 a(n) = Product_{k=0..n} (4*k)! / k!^4.

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A367569 a(n) = Product_{k=0..n} (5*k)! / k!^5.

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A367570 a(n) = Product_{k=0..n} (6*k)! / k!^6.

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A367571 a(n) = Product_{k=0..n} (7*k)! / k!^7.

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A324566 a(n) = Product_{i=0..n, j=0..n} (binomial(2*i, i) + binomial(2*j, j)).

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A338550 Number of binary trees of height n such that the number of nodes at depth d equals d+1 for every d = 0..n.

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A324567 a(n) = Product_{i=0..n, j=0..n} (binomial(2*i, j) + binomial(2*j, i)).

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A324568 a(n) = Sum_{i=0..n, j=0..n} (binomial(2*i, j) + binomial(2*j, i)).

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A371646 a(n) = Product_{k=0..n} binomial(n^3, k^3).

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A374891 Obverse convolution (1)**A000984; see Comments.

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A324566 a(n) = Product_{i=0..n, j=0..n} (binomial(2i, i) + binomial(2j, j)).

A324567 a(n) = Product_{i=0..n, j=0..n} (binomial(2i, j) + binomial(2j, i)).

A324568 a(n) = Sum_{i=0..n, j=0..n} (binomial(2i, j) + binomial(2j, i)).