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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A216616 G.f. satisfies: A(x) = (1 + x*(1-x)*A(x)) * (1 + x^2*A(x)^2).

Original entry on oeis.org

1, 1, 1, 3, 7, 15, 39, 103, 267, 719, 1975, 5447, 15199, 42863, 121647, 347455, 998559, 2884143, 8367599, 24377503, 71282351, 209132511, 615447711, 1816255583, 5373748287, 15937008575, 47368376255, 141075930495, 420957812863, 1258317356799, 3767538459391
Offset: 0

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Author

Paul D. Hanna, Sep 10 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 7*x^4 + 15*x^5 + 39*x^6 + 103*x^7 +...
The logarithm of the g.f. begins:
log(A(x)) = ((1-x) + x*A(x))*x + ((1-x)^2 + 2^2*x*(1-x)*A(x) + x^2*A(x)^2)*x^2/2 +
((1-x)^3 + 3^2*x*(1-x)^2*A(x) + 3^2*x^2*(1-x)*A(x)^2 + x^3*A(x)^3)*x^3/3 +
((1-x)^4 + 4^2*x*(1-x)^3*A(x) + 6^2*x^2*(1-x)^2*A(x)^2 + 4^2*x^3*(1-x)*A(x)^3 + x^4*A(x)^4)*x^4/4 +
((1-x)^5 + 5^2*x*(1-x)^4*A(x) + 10^2*x^2*(1-x)^3*A(x)^2 + 10^2*x^3*(1-x)^2*A(x)^3 + 5^2*x^4*(1-x)*A(x)^4 + x^5*A(x)^5)*x^5/5 +...
Explicitly,
log(A(x)) = x + x^2/2 + 7*x^3/3 + 17*x^4/4 + 41*x^5/5 + 133*x^6/6 + 393*x^7/7 + 1121*x^8/8 + 3373*x^9/9 + 10161*x^10/10 +...

Crossrefs

Cf. A216617, A216604, A216447.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=(1 + x*(1-x)*A)*(1+x^2*A^2) +x*O(x^n));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^2*x^k*(1-x)^(m-k)*A^k +x*O(x^n)))));polcoeff(A,n)}
for(n=0,40,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k) * A(x)^k ).
Recurrence: 2*(n+1)*(2*n+3)*(1367*n^5 - 22152*n^4 + 133091*n^3 - 342822*n^2 + 320852*n - 46056)*a(n) = (25973*n^7 - 386713*n^6 + 1995969*n^5 - 3489635*n^4 - 1128062*n^3 + 4916268*n^2 + 111144*n - 1105344)*a(n-1) - (42377*n^7 - 677143*n^6 + 3973061*n^5 - 9668137*n^4 + 6180338*n^3 + 7421072*n^2 - 9286128*n + 2230560)*a(n-2) + 4*(20505*n^7 - 365088*n^6 + 2506402*n^5 - 7967959*n^4 + 10412513*n^3 + 122671*n^2 - 10870392*n + 6017148)*a(n-3) - 2*(76552*n^7 - 1463333*n^6 + 10992560*n^5 - 39638307*n^4 + 64155864*n^3 - 18295816*n^2 - 55893000*n + 38191680)*a(n-4) + 2*(76552*n^7 - 1553555*n^6 + 12502721*n^5 - 49103379*n^4 + 89371647*n^3 - 37863250*n^2 - 72758304*n + 53344368)*a(n-5) - 4*(20505*n^7 - 443007*n^6 + 3803465*n^5 - 16027031*n^4 + 31715194*n^3 - 16244518*n^2 - 25294728*n + 19078920)*a(n-6) + 4*(n-6)*(6835*n^6 - 116228*n^5 + 733337*n^4 - 1976556*n^3 + 1537404*n^2 + 1530848*n - 1351680)*a(n-7) - 4*(n-7)*(n-6)*(1367*n^5 - 15317*n^4 + 58153*n^3 - 62791*n^2 - 47292*n + 44280)*a(n-8). - Vaclav Kotesovec, Dec 21 2013
a(n) ~ c*d^n/n^(3/2), where d = 3.14415377058430689... is the root of the equation -4 + 16*d - 44*d^2 + 68*d^3 - 44*d^4 + 16*d^5 - 15*d^6 + 4*d^7 = 0 and c = 0.77951549908443860621183... - Vaclav Kotesovec, Dec 21 2013
Constant c = 1/(8*sqrt(Pi*r)), where r = 0.008185036943737927662526644... is the root of the equation -5468 + 436073*r + 8837888*r^2 + 2268581888*r^3 - 4115660800*r^4 + 2178940928000*r^5 + 3543348019200*r^6 + 1717986918400*r^7 = 0. - Vaclav Kotesovec, Jan 04 2014

A346075 a(n) = 1 + Sum_{k=1..n-3} a(k) * a(n-k-3).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 6, 10, 16, 25, 41, 69, 115, 192, 326, 558, 955, 1641, 2839, 4930, 8578, 14972, 26222, 46037, 80988, 142793, 252307, 446617, 791885, 1406394, 2501642, 4456080, 7947963, 14194221, 25379751, 45430710, 81409233, 146028788, 262192876, 471193406
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 04 2021

Keywords

Links

Crossrefs

Cf. A105633, A216604, A217282, A307970, A343304, A346047, A346076, A346077.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[a[k] a[n - k - 3], {k, 1, n - 3}]; Table[a[n], {n, 0, 40}]
nmax = 40; A[] = 0; Do[A[x] = 1/(1 - x) + x^3 A[x] (A[x] - 1) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
  • SageMath
    @CachedFunction
def a(n): # a = A346075
    if (n<4): return 1
    else: return 1 + sum(a(k)*a(n-k-3) for k in range(1,n-2))
[a(n) for n in range(51)] # G. C. Greubel, Nov 27 2022

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x^3 * A(x) * (A(x) - 1).

A346503 G.f. A(x) satisfies A(x) = 1 + x^3 * A(x)^2 / (1 - x).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 5, 7, 14, 26, 43, 79, 148, 264, 483, 903, 1664, 3080, 5771, 10795, 20209, 38059, 71799, 135569, 256762, 487310, 925981, 1762841, 3361897, 6419595, 12275301, 23505143, 45061424, 86485016, 166176499, 319630115, 615387675, 1185940209, 2287527119, 4416083429
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 21 2021

Keywords

Crossrefs

Cf. A002212, A023426, A023431, A090345, A216604 (partial sums), A346504, A366588, A376490.

Programs

  • Mathematica
    nmax = 40; A[] = 0; Do[A[x] = 1 + x^3 A[x]^2/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 40}]

Formula

a(0) = 1, a(1) = a(2) = 0; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).
a(n) ~ 2^(n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 30 2021
From Seiichi Manyama, Sep 26 2024: (Start)
G.f.: 2/(1 + sqrt(1 - 4*x^3/(1 - x))).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(n-2*k-1,n-3*k) / (k+1). (End)

A364589 G.f. satisfies A(x) = 1/(1-x) + x^3*A(x)^3.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 14, 31, 67, 146, 331, 760, 1749, 4072, 9583, 22673, 53929, 129055, 310328, 749152, 1815481, 4415313, 10771564, 26352955, 64644926, 158963191, 391767016, 967523138, 2394060433, 5934576763, 14735792889, 36647185192, 91274339014, 227645446307
Offset: 0

Views

Author

Seiichi Manyama, Jul 29 2023

Keywords

Crossrefs

Cf. A216604, A226974.
Cf. A049130, A199475, A364590.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n-k, 2*k)*binomial(3*k,k)/(2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n-k,2*k) * binomial(3*k,k) / (2*k+1).
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