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)) ).

A385834 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k^6) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 200, 146401, 600098283, 9378336443140, 437583801957155730, 51482609496251191260549, 13496011632930307406903060651, 7172374406405634119759727327588155, 7172395923569361382696722735713532276498, 12706358411963754476880803069979932030145242780
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A088716, A385830, A385831, A385832, A385833.

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

Formula

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

A385843 a(n) = 1 + Sum_{k=0..n-1} k^5 * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 66, 16105, 16507753, 51603272051, 401318681776723, 6745364508844808841, 221038850400001766938953, 13052344129663319516736911260, 1305247465753403752473945799113276, 210212714880649951675343095297590137757, 52307860484508916277278208388919504757392477
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A143917, A385840, A385841, A385842.
Cf. A385833, A385838.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, j^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} Stirling2(5,k) * x^k * (d^k/dx^k A(x)) ) ).

A385942 a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k) * (1 + k^5) * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 5, 508, 497861, 2554041696, 47918955042217, 2608995595530944320, 350836859825187730934697, 103472315352121087796983183360, 61101436986101317921145771113951181, 67212924933426575369862458525709786073344, 129898118403746997254471428114728554653243564525
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2025

Keywords

Crossrefs

Cf. A156326, A385939, A385940, A385941, A385943.
Cf. A385833.

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^5)*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..5} Stirling2(5,k) * x^k * (d^k/dx^k A(x)) ).

A386446 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} k^5 * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 67, 16414, 16840826, 52661283276, 409599480216723, 6884957718009061046, 225620064835937122627934, 13323090455565480199133495252, 1332335691963961772604470940370302, 214576660211223693770379106296061734124, 53393968668333658608864584261609697870131860
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2025

Keywords

Crossrefs

Cf. A075834, A386443, A386444, A386445, A386447.
Cf. A385833.

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

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x - x*Sum_{k=1..5} Stirling2(5,k) * x^k * (d^k/dx^k A(x)) ).
Showing 1-7 of 7 results.

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