cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A201555 a(n) = C(2*n^2,n^2) = A000984(n^2), where A000984 is the central binomial coefficients.

Original entry on oeis.org

1, 2, 70, 48620, 601080390, 126410606437752, 442512540276836779204, 25477612258980856902730428600, 23951146041928082866135587776380551750, 365907784099042279561985786395502921046971688680, 90548514656103281165404177077484163874504589675413336841320
Offset: 0

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Author

Paul D. Hanna, Dec 02 2011

Keywords

Comments

Central coefficients of triangle A228832.

Examples

			L.g.f.: L(x) = 2*x + 70*x^2/2 + 48620*x^3/3 + 601080390*x^4/4 + ...
where exponentiation equals the g.f. of A201556:
exp(L(x)) = 1 + 2*x + 37*x^2 + 16278*x^3 + 150303194*x^4 + ... + A201556(n)*x^n + ...

Links

Crossrefs

Cf. A000984, A007814, A159918, A201556, A214441, A228832, A285388, A285717.

Programs

  • Mathematica
    Table[Binomial[2n^2,n^2],{n,0,10}] (* Harvey P. Dale, Dec 10 2011 *)
  • PARI
    a(n) = binomial(2*n^2,n^2)
  • Python
    from math import comb
def A201555(n): return comb((m:=n**2)<<1,m) # Chai Wah Wu, Jul 08 2022

Formula

L.g.f.: ignoring initial term, equals the logarithmic derivative of A201556.
a(n) = (2*n^2)! / (n^2)!^2.
a(n) = Sum_{k=0..n^2} binomial(n^2,k)^2.
For primes p >= 5: a(p) == 2 (mod p^3), Oblath, Corollary II; a(p) == binomial(2*p,p) (mod p^6) - see Mestrovic, Section 5, equation 31. - Peter Bala, Dec 28 2014
A007814(a(n)) = A159918(n). - Antti Karttunen, Apr 27 2017, based on Vladimir Shevelev's Jul 20 2009 formula in A000984.

A206830 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, n*k) * x^k ).

Original entry on oeis.org

1, 1, 2, 5, 34, 520, 14397, 993806, 222547738, 98753510701, 66772601607218, 82150206439975648, 310163020349941301606, 3022167582612808506550780, 47176617497043375266215814522, 1129578055293824008530028604347686, 62478430488069985838347598494293429802
Offset: 0

Views

Author

Paul D. Hanna, Feb 12 2012

Keywords

Comments

Note: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n^2} binomial(n^2, k) * x^k ) does not yield an integer series (see A227467).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 34*x^4 + 520*x^5 + 14397*x^6 + ...
such that, by definition, the logarithm equals:
log(A(x)) = x*(1+x) + x^2*(1 + 6*x + x^2)/2 + x^3*(1 + 84*x + 84*x^2 + x^3)/3 + x^4*(1 + 1820*x + 12870*x^2 + 1820*x^3 + x^4)/4 + x^5*(1 + 53130*x + 3268760*x^2 + 3268760*x^3 + 53130*x^4 + x^5)/5 + ... + x^n/n*Sum_{k=0..n} A209330(n,k)*x^k + ...
More explicitly,
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 115*x^4/4 + 2416*x^5/5 + 83064*x^6/6 + ...

Crossrefs

Cf. A167006, A201556, A227467, A209330, A207137, A228905.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m^2, m*k)*x^k)*x^m/m)+x*O(x^n)), n)}
for(n=0,15,print1(a(n),", "))

A224732 G.f.: exp( Sum_{n>=1} binomial(2*n,n)^n * x^n/n ).

Original entry on oeis.org

1, 2, 20, 2704, 6008032, 203263062688, 103724721990326528, 801185400238209125917312, 94088900962948953837864576996352, 168691065596220817138271126002845218561536, 4634314586972355372645450331391809316221983940020224
Offset: 0

Views

Author

Paul D. Hanna, Apr 16 2013

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 20*x^2 + 2704*x^3 + 6008032*x^4 + 203263062688*x^5 +...
where
log(A(x)) = 2*x + 6^2*x^2/2 + 20^3*x^3/3 + 70^4*x^4/4 + 252^5*x^5/5 + 924^6*x^6/6 + 3432^7*x^7/7 + 12870^8*x^8/8 +...+ A000984(n)^n*x^n/n +...

Links

Crossrefs

Cf. A200002, A224733, A201556, A224734, A224735, A224736, A000984.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(k=1,n,binomial(2*k,k)^k*x^k/k)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

Formula

Logarithmic derivative yields A224733.
a(n) ~ exp(-1/8) * 2^(2*n^2) / (Pi^(n/2) * n^(1 + n/2)). - Vaclav Kotesovec, Jan 26 2015
a(n) ~ (binomial(2*n,n))^n / n. - Vaclav Kotesovec, Jan 26 2015

A200002 G.f.: exp( Sum_{n>=1} C(2*n,n)^n/2^n * x^n/n ).

Original entry on oeis.org

1, 1, 5, 338, 375502, 6351970709, 1620698781098852, 6259260939361008796229, 367534769386519350929158503892, 329474737492618783473185792974307067503, 4525697838840190793599072589249813785373031191426, 955617474162634862818320009634143510233705849191099879550608
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2011

Keywords

Comments

Compare to the g.f. C(x) = 1 + x*C(x)^2 of the Catalan numbers (A000108), where C(x) = exp( Sum_{n>=1} C(2*n,n)/2 * x^n/n ).

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 338*x^3 + 375502*x^4 + 6351970709*x^5 +...
where
log(A(x)) = x + 3^2*x^2/2 + 10^3*x^3/3 + 35^4*x^4/4 + 126^5*x^5/5 + 462^6*x^6/6 + 1716^7*x^7/7 +...+ A001700(n+1)^n*x^n/n +...

Crossrefs

Cf. A224732, A201556, A001700.

Programs

  • Mathematica
    nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[2*k,k]^k/2^k * b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b  (* Vaclav Kotesovec, Mar 06 2014 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,binomial(2*m,m)^m/2^m*x^m/m)+x*O(x^n)),n)}

Formula

Conjecture: Limit n->infinity a(n)^(1/n^2) = 4. - Vaclav Kotesovec, Mar 06 2014

A213409 G.f.: exp( Sum_{n>=1} binomial(3*n^2,n^2) * x^n/n ).

Original entry on oeis.org

1, 3, 252, 1563022, 563716946982, 10517711119760250261, 9692061982207456039533424586, 430311348543725825536505706371595438684, 906895928239445077568583988067142630846220290783969, 89857639488565787203362892584824012528872539028234934088960440084
Offset: 0

Views

Author

Paul D. Hanna, Jun 10 2012

Keywords

Comments

Compare to the g.f. G(x) = 1 + x*G(x)^3 of A001764: G(x)^3 = exp( Sum_{n>=1} binomial(3*n,n) * x^n/n ).

Examples

			G.f.: A(x) = 1 + 3*x + 252*x^2 + 1563022*x^3 + 563716946982*x^4 +...
where
log(A(x)) = 3*x + 495*x^2/2 + 4686825*x^3/3 + 2254848913647*x^4/4 + 52588547141148893628*x^5/5 +...+ C(3*n^2,n^2)*x^n/n +...

Links

Crossrefs

Cf. A201556, A213410, A001764, A245245 (log).

Programs

  • Mathematica
    nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[3*k^2, k^2]*b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b  (* Vaclav Kotesovec, Mar 06 2014 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, binomial(3*m^2, m^2)*x^m/m)+x*O(x^n)), n)}
  • PARI
    {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, binomial(3*k^2, k^2)*a(n-k)))}

Formula

a(n) = (1/n) * Sum_{k=1..n} binomial(3*k^2,k^2) * a(n-k) for n>0 with a(0)=1.
a(n) ~ sqrt(3) * (27/4)^(n^2) / (2*sqrt(Pi)*n^2). - Vaclav Kotesovec, Mar 06 2014

A213402 Expansion of exp( Sum_{n>=1} binomial(2*n^2-1, n^2) * x^n/n ).

Original entry on oeis.org

1, 1, 18, 8121, 75143314, 12641135924840, 36876057665515782388, 1819829483946334379705083199, 1496946629440335290077827171482478552, 20328210229221517747154723779733726877327901902, 4527425732825492268524878479039840102127714838348518156649
Offset: 0

Views

Author

Paul D. Hanna, Jun 10 2012

Keywords

Comments

Self-convolution square-root of A201556.
Compare to the g.f. C(x) = 1 + x*C(x)^2 of the Catalan numbers (A000108): C(x) = exp( Sum_{n>=1} binomial(2*n-1,n) * x^n/n ).

Examples

			G.f.: A(x) = 1 + x + 18*x^2 + 8121*x^3 + 75143314*x^4 + 12641135924840*x^5 +...
where
log(A(x)) = x + 35*x^2/2 + 24310*x^3/3 + 300540195*x^4/4 +...+ C(2*n^2-1,n^2)*x^n/n +...

Links

Crossrefs

Cf. A201556, A200002.

Programs

  • Mathematica
    nmax = 10; b = ConstantArray[0, nmax+1]; b[[1]] = 1; Do[b[[n+1]] = 1/n*Sum[Binomial[2*k^2, k^2]/2*b[[n-k+1]], {k, 1, n}], {n, 1, nmax}]; b  (* Vaclav Kotesovec, Mar 06 2014 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n, binomial(2*m^2, m^2)/2*x^m/m)+x*O(x^n)), n)}
  • PARI
    {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, binomial(2*k^2, k^2)/2*a(n-k)))}
for(n=0,21,print1(a(n),", "))

Formula

a(n) = (1/n) * Sum_{k=1..n} C(2*k^2,k^2)/2 * a(n-k) for n>0 with a(0)=1.
a(n) ~ 2^(2*n^2-1) / (sqrt(Pi)*n^2). - Vaclav Kotesovec, Mar 06 2014
Showing 1-6 of 6 results.

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