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.

A382029 E.g.f. A(x) satisfies A(x) = exp(x*C(x*A(x)^2)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 1, 3, 31, 529, 12601, 385891, 14440567, 638576065, 32580927505, 1883889232291, 121742057314351, 8695278706372369, 680187946863332233, 57833833258995140803, 5310742450917819399751, 523793286672328763358721, 55223769332070053104438945, 6197871354601209094032190147
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A212722, A382030, A382031.
Cf. A161629, A382042.
Cf. A000108, A214688, A214689, A379690.

Programs

  • PARI
    a(n) = if(n==0, 1, n!*sum(k=0, n-1, (2*k+1)^(n-k-1)*binomial(n+k, k)/((n+k)*(n-k-1)!)));

Formula

Let F(x) be the e.g.f. of A379690. F(x) = log(A(x))/x = C(x*A(x)^2).
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(2*x)) ) ).
a(n) = n! * Sum_{k=0..n-1} (2*k+1)^(n-k-1) * binomial(n+k,k)/((n+k) * (n-k-1)!) for n > 0.

A379689 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(x) * (1 - x*exp(2*x)) ).

Original entry on oeis.org

1, 0, 5, 26, 557, 9504, 254737, 7405712, 264468185, 10599167744, 484155176381, 24530813822976, 1373346539948869, 83980088153710592, 5576376312266516681, 399370804845913339904, 30695207044654060184753, 2519882221014204064727040, 220076205166821624927515893
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2024

Keywords

Crossrefs

Cf. A379661, A379690, A379691.

Programs

  • PARI
    a(n) = sum(k=0, n, (n-2*k-1)^k * (2*n-k)!/(k!*(n-k)!))/(n+1);

Formula

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

A379691 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) * (1 - x*exp(2*x)) ).

Original entry on oeis.org

1, 2, 17, 274, 6597, 212736, 8624581, 421843472, 24185705417, 1591194859264, 118184516071641, 9782950785024000, 893132377427288653, 89156432069922504704, 9661304014254414999821, 1129505503357457643206656, 141711496280128816909982097, 18992404410135723679211716608
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2024

Keywords

Crossrefs

Cf. A379661, A379689, A379690.

Programs

  • PARI
    a(n) = sum(k=0, n, (3*n-2*k+1)^k * (2*n-k)!/(k!*(n-k)!))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} (3*n-2*k+1)^k * (2*n-k)!/(k! * (n-k)!).

A382039 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(3*x)) ).

Original entry on oeis.org

1, 1, 10, 147, 3252, 96165, 3569778, 159771717, 8378589096, 504057519945, 34227869887710, 2589957885708369, 216121694333055228, 19717935804239270013, 1952741002119283320714, 208629930642065967641805, 23919711023929511941080912, 2929406351866509691077727761
Offset: 0

Views

Author

Seiichi Manyama, Mar 13 2025

Keywords

Crossrefs

Cf. A213644, A379690, A382040.
Cf. A366233.

Programs

  • PARI
    a(n) = sum(k=0, n, (3*k)^(n-k)*(n+k)!/(k!*(n-k)!))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^2*exp(3*x*A(x)).
a(n) = (1/(n+1)) * Sum_{k=0..n} (3*k)^(n-k) * (n+k)!/(k! * (n-k)!).

A382043 E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^3*exp(2*x*A(x)).

Original entry on oeis.org

1, 1, 10, 168, 4280, 146840, 6354432, 332467072, 20419261312, 1440559380096, 114820434103040, 10205253450850304, 1000815286620229632, 107355373421379825664, 12504295470535952613376, 1571670041412254073323520, 212035122185327799251468288, 30561822671438790519426154496
Offset: 0

Views

Author

Seiichi Manyama, Mar 13 2025

Keywords

Crossrefs

Cf. A366232, A379690, A382044.
Cf. A364984, A382030.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (2*k)^(n-k)*binomial(n+2*k+1, k)/((n+2*k+1)*(n-k)!));

Formula

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

A382044 E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^4*exp(2*x*A(x)).

Original entry on oeis.org

1, 1, 12, 252, 8096, 352120, 19372512, 1290832480, 101078857728, 9098805892608, 925857411706880, 105098610198360064, 13167689873652178944, 1804954814456584081408, 268702350796640969736192, 43172786067215188056023040, 7446421094705349321120677888, 1372319952106065844255081037824
Offset: 0

Views

Author

Seiichi Manyama, Mar 13 2025

Keywords

Crossrefs

Cf. A366232, A379690, A382043.
Cf. A365175, A382031.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (2*k)^(n-k)*binomial(n+3*k+1, k)/((n+3*k+1)*(n-k)!));

Formula

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

A382040 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(4*x)) ).

Original entry on oeis.org

1, 1, 12, 198, 4912, 163120, 6796224, 341366704, 20088997632, 1356164492544, 103333898644480, 8773563043734016, 821474949840482304, 84093840447771701248, 9344359942839980900352, 1120159940123276849141760, 144096985208727744665288704, 19800296439825918648654561280
Offset: 0

Views

Author

Seiichi Manyama, Mar 13 2025

Keywords

Crossrefs

Cf. A213644, A379690, A382039.
Cf. A366234, A382031.

Programs

  • PARI
    a(n) = sum(k=0, n, (4*k)^(n-k)*(n+k)!/(k!*(n-k)!))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^2*exp(4*x*A(x)).
a(n) = (1/(n+1)) * Sum_{k=0..n} (4*k)^(n-k) * (n+k)!/(k! * (n-k)!).
Showing 1-7 of 7 results.

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