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 1-7 of 7 results.

A385830 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k^2) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 20, 241, 4623, 130300, 5100750, 265780029, 17827454651, 1498498011875, 154408489507578, 19151761451917580, 2815820822235814540, 484383420815495253624, 96401320782466194458886, 21981036279413999807199045, 5693391431445001330242504699, 1662538953499888924638316487305
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Links

Crossrefs

Cf. A088716, A385831, A385832, A385833, A385834.
Cf. A385762, A385835.

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, (1+j^2)*v[j+1]*v[i-j])); v;

Formula

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

A385833 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k^5) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 104, 25585, 26276091, 82191698776, 639369308538270, 10747798328839679301, 352216100969784522738455, 20799065226839989441184616755, 2079968920938449464603267217930862, 334987314655287149221766445992266495796, 83356568448492338030736248231384628286761124
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A088716, A385830, A385831, A385832, A385834.
Cf. A385765, A385843.

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

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) - x*Sum_{k=1..5} Stirling2(5,k) * x^k * (d^k/dx^k A(x)) ).

A385831 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k^3) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 32, 961, 64467, 8255248, 1808137854, 625644428013, 322212826476551, 235861774406899499, 236570361788785389414, 315585587694401993913716, 546279374467805677562555764, 1201815582876341559500261276952, 3301389061225358326490572037897646
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A088716, A385830, A385832, A385833, A385834.
Cf. A385763.

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

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) - x*Sum_{k=1..3} Stirling2(3,k) * x^k * (d^k/dx^k A(x)) ).

A385832 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k^4) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 56, 4705, 1218747, 765389596, 994245193386, 2390167881074445, 9797301213263859467, 64309492440202351088387, 643287882516349276270085850, 9420307945482704895570131173916, 195367768417628005309741727943311572, 5580484965405704420901774303244279908840
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A088716, A385830, A385831, A385833, A385834.
Cf. A385764.

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

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) - x*Sum_{k=1..4} Stirling2(4,k) * x^k * (d^k/dx^k A(x)) ).

A385943 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k) * (1 + k^6) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 5, 988, 2888933, 59194266336, 5550172939486537, 1812719786900514856960, 1706146365658760367161728617, 4025335006744077207541517795929600, 21392361120121469487882204135345762936461, 235316442953945260569915546964215106936729204224
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A156326, A385939, A385940, A385941, A385942.
Cf. A385834.

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

Formula

E.g.f. A(x) satisfies A(x) = exp( x*A(x) + x*Sum_{k=1..6} Stirling2(6,k) * x^k * (d^k/dx^k A(x)) ).

A386447 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} k^6 * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 131, 95760, 392424606, 6132419429842, 286126426174265119, 33663060172069656177612, 8824636572155130972996888814, 4689791333849576329442118802082252, 4689800713441077274969296364554337253614, 8308277421310507219950890075481144453543272228
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2025

Keywords

Crossrefs

Cf. A075834, A386443, A386444, A386445, A386446.
Cf. A385834.

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

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x - x*Sum_{k=1..6} Stirling2(6,k) * x^k * (d^k/dx^k A(x)) ).

A385839 a(n) = 1 + Sum_{k=0..n-1} (1 + k^6) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 2, 7, 471, 345240, 1415486250, 22122636527386, 1032242227753172079, 121446394933841583123508, 31836929544298684420302348229, 16919577022277987344334514604394117, 16919644700745370569015746375165719379327, 29974250364360598877961318618919670090162246645
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A321087, A385835, A385836, A385837, A385838.
Cf. A385834.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, (1+j^6)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x*A(x) - x*Sum_{k=1..6} Stirling2(6,k) * x^k * (d^k/dx^k A(x)) ) ).
Showing 1-7 of 7 results.

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