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 1-6 of 6 results.

A385945 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(k+3,3) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 5, 63, 1533, 62736, 3969387, 366744330, 47441881377, 8313978813120, 1921417594566561, 572533956456137424, 215766174031503450885, 101144655173329674617088, 58127411808811103704523775, 40435528907318329027426583376, 33666103690446265067517343384833
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A000272, A156325, A385946, A385947, A385948.
Cf. A385952.

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

Formula

E.g.f. A(x) satisfies A(x) = exp( Sum_{k=0..2} binomial(2,k) * x^(k+1)/(k+1)! * (d^k/dx^k A(x)) ), where (d^0/dx^0 A(x)) = A(x) by convention.

A385947 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(k+5,5) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 7, 166, 10029, 1321025, 341733205, 160453080950, 128422430092385, 166469443066352440, 334968718604910165425, 1009644894131844004090200, 4422360688027934597152329025, 27423466157672001507611296316100, 235350249980804930971638499216115775
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A000272, A156325, A385945, A385946, A385948.
Cf. A385954.

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

Formula

E.g.f. A(x) satisfies A(x) = exp( Sum_{k=0..4} binomial(4,k) * x^(k+1)/(k+1)! * (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.

A386512 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * binomial(k+3,4) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 52, 3325, 598906, 255199051, 226888865575, 382997189880593, 1140957869006770561, 5659169551911928576531, 44571684957086887771692731, 535930324156886354251195391269, 9517054240482595566592327616630965, 242627830243798770154326313268171970697
Offset: 0

Views

Author

Seiichi Manyama, Jul 24 2025

Keywords

Crossrefs

Cf. A143925, A386510, A386511, A386513.
Cf. A385946, A386454.

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, (1+j)*binomial(j+3, 4)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

E.g.f. A(x) satisfies A(x) = exp( x + x*Sum_{k=1..4} binomial(3,k-1) * x^k/k! * (d^k/dx^k A(x)) ).

A385980 a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * binomial(k+3,3) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 9, 295, 24921, 4504516, 1543745107, 919392117722, 890353538984905, 1330464112593541120, 2940642877993896450701, 9284167814032856189142864, 40666099850492306669400356041, 241073945237343019120798232332320, 1893421587381601800604423881821405775
Offset: 0

Views

Author

Seiichi Manyama, Jul 14 2025

Keywords

Crossrefs

Cf. A000272, A156326, A385979, A385981, A385982.
Cf. A385946, A385952.

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

Formula

E.g.f. A(x) satisfies A(x) = exp( 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.
Showing 1-6 of 6 results.

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

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