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.

Previous Showing 21-29 of 29 results.

A380096 E.g.f. A(x) satisfies A(x) = 1/( 1 - 3*x*A(x)^3*exp(x*A(x)^3) )^(1/3).

Original entry on oeis.org

1, 1, 12, 289, 10724, 540745, 34551886, 2676439507, 243782162408, 25535467766593, 3024360522754010, 399665508962874451, 58301379215119084012, 9305724270031402836337, 1613262216112899513140630, 301870732625016111841693795, 60639884085040694650040518736
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2025

Keywords

Crossrefs

Cf. A213644, A380095.
Cf. A380043, A380097.

Programs

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

Formula

E.g.f.: ( (1/x) * Series_Reversion(x*(1 - 3*x*exp(x))) )^(1/3).
a(n) = n! * Sum_{k=0..n} 3^k * k^(n-k) * binomial(n+k+1/3,k)/( (3*n+3*k+1)*(n-k)! ).
a(n) = (n!/(3*n+1)) * Sum_{k=0..n} (-3)^k * k^(n-k) * binomial(-n-1/3,k)/(n-k)!.

A380097 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - 3*x*exp(x)) ).

Original entry on oeis.org

1, 3, 42, 1089, 42132, 2182335, 142084818, 11159447943, 1027313395944, 108517938075387, 12940759400071710, 1719811206219287643, 252076045285741340700, 40398758175398949144039, 7028240082095865121961514, 1319141702032289451776382975, 265703833060229155917857703888
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2025

Keywords

Crossrefs

Cf. A213644, A379688.
Cf. A380096.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = 1/( 1 - 3*x*A(x)*exp(x*A(x)) ).
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A380096.
a(n) = (1/(n+1)) * Sum_{k=0..n} 3^k * k^(n-k) * (n+k)!/(k! * (n-k)!).
a(n) == 0 (mod 3) for n>0.

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)!).

A370986 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^2/2*exp(x)) ).

Original entry on oeis.org

1, 0, 1, 3, 24, 220, 2535, 35931, 588028, 11110212, 236355885, 5605425595, 146586048906, 4190560545678, 130037568893323, 4352949788266275, 156362710268877960, 5999465853656510056, 244887352806333261753, 10595948826118665719475, 484448170190529051468910
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A213644, A370987.
Cf. A346888, A370930.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x^2/2*exp(x)))/x))
  • PARI
    a(n) = sum(k=0, n\2, k^(n-2*k)*(n+k)!/(2^k*k!*(n-2*k)!))/(n+1);

Formula

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

A370987 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^3/6*exp(x)) ).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 100, 1295, 11256, 110964, 1713720, 27147285, 409295260, 7192099486, 146953847404, 3083283003255, 67579239087920, 1632183291677800, 42282787050941616, 1140103579166365929, 32452333261311639540, 982975683671186407090, 31244119064026146137860
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A213644, A370986.
Cf. A346889, A370931.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x^3/6*exp(x)))/x))
  • PARI
    a(n) = sum(k=0, n\3, k^(n-3*k)*(n+k)!/(6^k*k!*(n-3*k)!))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} k^(n-3*k) * (n+k)!/(6^k * k! * (n-3*k)!).

A379685 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x) * (1 - x*exp(x)) ).

Original entry on oeis.org

1, 3, 30, 551, 15028, 547717, 25068058, 1383323517, 89443699176, 6634682537993, 555501170856934, 51828125728865257, 5332620999430989244, 599894268098223894525, 73253745510185331985842, 9650159930850877102454693, 1364228585624978795929566928, 206008264557747708717576118417
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2024

Keywords

Crossrefs

Cf. A213644, A377890, A379684.

Programs

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

Formula

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

A380077 Expansion of e.g.f. (1/x) * Series_Reversion( x * sqrt(1 - 2*x*exp(x)) ).

Original entry on oeis.org

1, 1, 7, 87, 1621, 40485, 1271841, 48220207, 2143450009, 109350344745, 6298638659245, 404371344546411, 28633701543626037, 2217105596852342989, 186362307297569836993, 16901012222196104542695, 1644911203243501609414321, 171017059743998995011125457, 18916512667390427993433246357
Offset: 0

Views

Author

Seiichi Manyama, Jan 11 2025

Keywords

Crossrefs

Cf. A213644, A380078.
Cf. A380035.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*sqrt(1-2*x*exp(x)))/x))
  • PARI
    a(n) = n!*sum(k=0, n, 2^k*k^(n-k)*binomial(n/2+k+1/2, k)/((n+2*k+1)*(n-k)!));

Formula

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

A380078 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - 3*x*exp(x))^(1/3) ).

Original entry on oeis.org

1, 1, 8, 115, 2484, 72005, 2626846, 115688349, 5974568552, 354154378249, 23704428986010, 1768459611322481, 145525743200753356, 13095070459815108141, 1279226572751177845718, 134827003107939467441845, 15250595677663579282034256, 1842758049329907303778372625
Offset: 0

Views

Author

Seiichi Manyama, Jan 11 2025

Keywords

Crossrefs

Cf. A213644, A380077.
Cf. A380039.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-3*x*exp(x))^(1/3))/x))
  • PARI
    a(n) = n!*sum(k=0, n, 3^k*k^(n-k)*binomial(n/3+k+1/3, k)/((n+3*k+1)*(n-k)!));

Formula

E.g.f. A(x) satisfies A(x) = 1/( 1 - 3*x*A(x)*exp(x*A(x)) )^(1/3).
a(n) = n! * Sum_{k=0..n} 3^k * k^(n-k) * binomial(n/3+k+1/3,k)/( (n+3*k+1)*(n-k)! ).
a(n) = (n!/(n+1)) * Sum_{k=0..n} (-3)^k * k^(n-k) * binomial(-n/3-1/3,k)/(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)!).
Previous Showing 21-29 of 29 results.

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

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