A324566 -id:A324566 - cp's OEIS frontend

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

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))).

A324565 a(n) = Product_{i=0..n, j=0..n} (C(i)+C(j)), where C(k) = Catalan(k) = A000108(k).

Original entry on oeis.org

2, 16, 5184, 3292047360, 431257150704844800000, 1660988174371634812975670031600844800000, 29842277901990912803108442836098281105491323166448779304684748800000

Offset: 0

Vaclav Kotesovec, Mar 07 2019

nonn

Cf. A000108, A003046, A324566.

Table[Product[CatalanNumber[i] + CatalanNumber[j], {i, 0, n}, {j, 0, n}], {n, 0, 7}]

a(n) ~ c * d^n * 2^(4*n^3/3 + 3*n^2) * exp(3*n^2/4) / (Pi^(n^2/2) * n^(3*n^2/2 + 3*n - alfa)), where
d = 1.755243767008515125359629831741161625491476427596217377285488419728083846...
alfa = 0.033913207252709141540468528132950860454628430850651543265936300855299...
c = 1.31959917222402005903928675604752142566298009509...

cp's OEIS Frontend

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

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Formula

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

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Mathematica

Formula

A324565 a(n) = Product_{i=0..n, j=0..n} (C(i)+C(j)), where C(k) = Catalan(k) = A000108(k).

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Author

Keywords

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A324565 a(n) = Product_{i=0..n, j=0..n} (C(i)+C(j)), where C(k) = Catalan(k) = A000108(k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

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)).