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 1-4 of 4 results.

A385874 a(n) = 1 + Sum_{k=0..n-1} binomial(k+1,2) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 8, 57, 639, 10357, 229588, 6686619, 248013315, 11425386222, 640413284553, 42933889931191, 3393203732253145, 312268381507616935, 33107736233111305459, 4006699123399932333697, 548987463226205098599755, 84552444466155546810368421, 14544161652321384236939516147
Offset: 0

Views

Author

Seiichi Manyama, Jul 11 2025

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, binomial(j+1, 2)*v[j+1]*v[i-j])); v;

Formula

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

A385875 a(n) = 1 + Sum_{k=0..n-1} binomial(k+2,3) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 10, 111, 2347, 84757, 4837213, 411373408, 49787445476, 8265626303452, 1826809978098228, 524311794034090050, 191377585766768936606, 87269255118865044728501, 48958442598180565027265909, 33340876732769115354996751746, 27239595466972699678481509900786
Offset: 0

Views

Author

Seiichi Manyama, Jul 11 2025

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, binomial(j+2, 3)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x*Sum_{k=1..3} binomial(2,k-1) * x^k/k! * (d^k/dx^k A(x)) ) ).

A385877 a(n) = 1 + Sum_{k=0..n-1} binomial(k+4,5) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 14, 309, 17637, 2240632, 566921596, 262489646519, 208155482551991, 268104800528280951, 537014337938584568385, 1613191612128443060280697, 7048035233444754041436840277, 43620293298146615746333469478901, 373782307403691698916363133787269075
Offset: 0

Views

Author

Seiichi Manyama, Jul 11 2025

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, binomial(j+4, 5)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x*Sum_{k=1..5} binomial(4,k-1) * x^k/k! * (d^k/dx^k A(x)) ) ).

A386454 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} binomial(k+3,4) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 13, 220, 8148, 586948, 75141039, 15930666825, 5289069956220, 2628685323745449, 1884772989271329869, 1890430039448133854031, 2584219798288871040676608, 4708450397910844142927823544, 11215531466814325127916787062534, 34341962107081618846057340207455738
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2025

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, binomial(j+3, 4)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x - x*Sum_{k=1..4} binomial(3,k-1) * x^k/k! * (d^k/dx^k A(x)) ).
Showing 1-4 of 4 results.