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 1-5 of 5 results.

A143568 E.g.f. satisfies A(x) = exp(x*A(x^4/4!)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 31, 106, 281, 946, 7561, 54286, 281161, 1207636, 7997991, 81996916, 701522641, 4580581916, 29742355441, 306369616636, 3632198902321, 34977922146721, 282526761829621, 2720464688299821, 36188717552636881, 464906756446099276, 4985291127563074901
Offset: 0

Views

Author

Alois P. Heinz, Aug 24 2008

Keywords

Links

Crossrefs

4th column of A143565.
Cf. A367720.

Programs

  • Maple
    A:= proc(n) option remember; if n<=0 then 1 else
      unapply(convert(series(exp(x*A(n-4)(x^4/24)), x, n+1), polynom), x) fi
    end:
a:= n-> coeff(A(n)(x), x,n)*n!:
seq(a(n), n=0..30);
  • Mathematica
    A[n_] := A[n] = If[n <= 0, 1&, Function[Normal[Series[Exp[y*A[n-2][y^4/4!]], {y, 0, n+1}] /. y -> #]]]; a[n_] := Coefficient[A[n][x], x, n]*n!; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 13 2014, after Maple *)

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/4)} (4*k+1) * a(k) * a(n-1-4*k) / (24^k * k! * (n-1-4*k)!). - Seiichi Manyama, Nov 28 2023

A367719 E.g.f. satisfies A(x) = exp(x*A(x^3)).

Original entry on oeis.org

1, 1, 1, 1, 25, 121, 361, 3361, 42001, 275185, 1819441, 30777121, 371238121, 9284332201, 131442054745, 1454933712961, 34120902859681, 851562584890081, 12300037440760801, 187928965721651905, 6019555345508794681, 130768735411230580441
Offset: 0

Views

Author

Seiichi Manyama, Nov 28 2023

Keywords

Comments

This sequence is different from A354553.

Crossrefs

Cf. A138292, A367720.
Cf. A354553.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=0, (i-1)\3, (3*j+1)*v[j+1]*v[i-3*j]/(j!*(i-1-3*j)!))); v;

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/3)} (3*k+1) * a(k) * a(n-1-3*k) / (k! * (n-1-3*k)!).

A367749 E.g.f. satisfies A(x) = exp(x * (1 + x + x^2 + x^3) * A(x^4)).

Original entry on oeis.org

1, 1, 3, 13, 73, 501, 4051, 37633, 394353, 4777993, 62569891, 893927541, 13827333433, 234241234813, 4212828738483, 80727388033321, 1641227208417121, 35581993575319953, 810641581182744643, 19416795485684156893, 487647253209539939241
Offset: 0

Views

Author

Seiichi Manyama, Nov 29 2023

Keywords

Crossrefs

Cf. A367747, A367748, A367751.
Cf. A367653, A367720.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=0, i-1, (j+1)*v[j\4+1]*v[i-j]/((j\4)!*(i-1-j)!))); v;

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..n-1} (k+1) * a(floor(k/4)) * a(n-1-k) / (floor(k/4)! * (n-1-k)!).

A367794 G.f. A(x) satisfies A(x) = 1 / (1 - x * A(x^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, 36, 50, 69, 95, 131, 181, 250, 346, 478, 660, 911, 1259, 1740, 2404, 3320, 4586, 6336, 8754, 12093, 16705, 23077, 31881, 44043, 60844, 84053, 116116, 160410, 221602, 306136, 422916, 584242, 807110, 1114996
Offset: 0

Views

Author

Seiichi Manyama, Nov 30 2023

Keywords

Links

Crossrefs

Cf. A000108, A000621, A367637, A367800.
Cf. A367653, A367692, A367694, A367720.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, (i-1)\4, v[j+1]*v[i-4*j])); v;
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A367794(n): return sum(A367794(k)*A367794(n-1-(k<<2)) for k in range(n+3>>2)) if n else 1 # Chai Wah Wu, Nov 30 2023

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} a(k) * a(n-1-4*k).

A367723 E.g.f. satisfies A(x) = exp(x*A(-x^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, -119, -719, -2519, -6719, 166321, 3598561, 29882161, 159572161, -389343239, -55939643759, -974399385959, -9282412863359, -46891283580959, 1814094098389441, 67045782535457761, 1076141148146824321, 61735522719009663721, 1058382395842664859121
Offset: 0

Views

Author

Seiichi Manyama, Nov 28 2023

Keywords

Crossrefs

Cf. A141369, A367721, A367722.
Cf. A367720.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=0, (i-1)\4, (-1)^j*(4*j+1)*v[j+1]*v[i-4*j]/(j!*(i-1-4*j)!))); v;

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/4)} (-1)^k * (4*k+1) * a(k) * a(n-1-4*k) / (k! * (n-1-4*k)!).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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