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.

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A370326 E.g.f.: exp(Sum_{k>=1} binomial(2*k,k) * x^k).

Original entry on oeis.org

1, 2, 16, 200, 3376, 71552, 1822144, 54131072, 1836436480, 70016026112, 2962490758144, 137711245058048, 6974788150104064, 382232015239454720, 22531888624878813184, 1421482338801856053248, 95553266255536369893376, 6817598649041309962600448, 514534725049116493981941760
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 15 2024

Keywords

Crossrefs

Cf. A000984, A065866, A213507, A251568, A370327.

Programs

  • Mathematica
    CoefficientList[Series[Exp[Sum[Binomial[2*k,k]*x^k, {k, 1, 20}]], {x, 0, 20}], x] * Range[0, 20]!
CoefficientList[Series[Exp[1/Sqrt[1 - 4*x] - 1], {x, 0, 20}], x] * Range[0, 20]!

Formula

E.g.f.: exp(1/sqrt(1 - 4*x) - 1).
a(n) ~ exp(3*n^(1/3)/2^(2/3) - n - 1) * 2^(2*n + 1/6) * n^(n - 1/3) / sqrt(3).

A382101 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(B_k(x) - 1), where B_k(x) = 1 + x*B_k(x)^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 13, 1, 1, 1, 7, 43, 73, 1, 1, 1, 9, 91, 529, 501, 1, 1, 1, 11, 157, 1753, 8501, 4051, 1, 1, 1, 13, 241, 4129, 45001, 169021, 37633, 1, 1, 1, 15, 343, 8041, 146001, 1447471, 4010455, 394353, 1, 1, 1, 17, 463, 13873, 362501, 6502681, 56041987, 110676833, 4596553, 1
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2025

Keywords

Examples

			Square array begins:
  1,   1,    1,     1,      1,      1, ...
  1,   1,    1,     1,      1,      1, ...
  1,   3,    5,     7,      9,     11, ...
  1,  13,   43,    91,    157,    241, ...
  1,  73,  529,  1753,   4129,   8041, ...
  1, 501, 8501, 45001, 146001, 362501, ...

Links

Crossrefs

Columns k=0..4 give A000012, A000262, A251568, A380512, A380516.
Cf. A355262, A382100.

Programs

  • PARI
    a(n, k) = if(n==0, 1, (n-1)!*pollaguerre(n-1, (k-1)*n+1, -1));

Formula

A(n,k) = (n-1)! * Sum_{j=0..n-1} binomial(k*n,j)/(n-j-1)! for n > 0.
A(n,k) = (n-1)! * LaguerreL(n-1, (k-1)*n+1, -1) for n > 0.
E.g.f. of column k: exp( Series_Reversion( x/(1+x)^k ) ).

A250916 E.g.f.: exp(C(x)^2 - 1) where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers, A000108.

Original entry on oeis.org

1, 2, 14, 152, 2236, 41512, 930904, 24474368, 738241424, 25132379552, 953267419744, 39867845243008, 1822779782497216, 90453927667906688, 4842249786763758464, 278167945047964156928, 17069371221016503644416, 1114374972408995525243392, 77126208846034435924819456
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2014

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x + 14*x^2/2! + 152*x^3/3! + 2236*x^4/4! + 41512*x^5/5! +...
such that log(A(x)) = C(x)^2 - 1,
log(A(x)) = 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 + 1430*x^7 +...
where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Crossrefs

Cf. A251568, A250917, A000108.

Programs

  • PARI
    {a(n)=local(C=1); for(i=0, n, C=1+x*C^2 +x*O(x^n)); n!*polcoeff(exp(C^2-1), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n) = n!*polcoeff(exp((1-2*x - sqrt(1-4*x + x^3*O(x^n)))/(2*x^2) - 1), n)}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f.: exp( (1-2*x-2*x^2 - sqrt(1-4*x))/(2*x^2) ).
a(n) ~ 2^(2*n+5/2) * n^(n-1) / exp(n-3). - Vaclav Kotesovec, Aug 22 2017
Previous Showing 11-13 of 13 results.

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