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.

A385952 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(k+3,3) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 5, 59, 1309, 48790, 2840931, 244770680, 29887602613, 4993307581843, 1108754325139526, 319359741512132370, 116893982001130825135, 53422902443413341967604, 30024521959524315980717288, 20477109546794819263709728560, 16750490995674468051531269811269
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A088716, A351798, A385953, A385954, A385955.
Cf. A385945.

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, binomial(j+3, 3)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - Sum_{k=0..3} binomial(3,k) * x^(k+1)/k! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.

A385948 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(k+6,6) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 8, 246, 21750, 4689546, 2197062708, 2046202234224, 3528088593902364, 10627093734265740672, 53295889303479275834616, 427383379745842299684115608, 5294446934064450139154214169992, 98355143996083993836475641916586304, 2669951662594756888115675117287929721248
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A000272, A156325, A385945, A385946, A385947.
Cf. A385955.

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, binomial(j+6, 6)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

E.g.f. A(x) satisfies A(x) = exp( Sum_{k=0..5} binomial(5,k) * x^(k+1)/(k+1)! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.

A385953 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(k+4,4) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 6, 101, 3756, 271256, 34761512, 7372486163, 2448035959989, 1216747945481685, 872431867857009866, 875060598719254613963, 1196215918953589596769516, 2179513438308809548333358500, 5191611931593198935913809439220, 15896735560092998091331427433546666
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A088716, A351798, A385952, A385954, A385955.
Cf. A385946.

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, binomial(j+4, 4)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - Sum_{k=0..4} binomial(4,k) * x^(k+1)/k! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.

A385954 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(k+5,5) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 7, 160, 9309, 1193192, 303192604, 140697031749, 111717191583621, 144005113804578040, 288587523313304535136, 867207126292422956078756, 3789698359352103250842742098, 23458242467926487526255374709015, 201037179886862036121457727887328687
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A088716, A351798, A385952, A385953, A385955.
Cf. A385947.

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, binomial(j+5, 5)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - Sum_{k=0..5} binomial(5,k) * x^(k+1)/k! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.
Showing 1-4 of 4 results.

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

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