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 1-10 of 12 results. Next

A380015 Expansion of e.g.f. 1/sqrt(1 - 2*x*exp(x)).

Original entry on oeis.org

1, 1, 5, 36, 361, 4640, 72771, 1347598, 28778849, 696288888, 18823644595, 562350743306, 18397666000209, 654164843763340, 25118967828553067, 1035914449832324070, 45665488606439586241, 2142825945301659242576, 106641225471890568771747, 5610282675990428302440130
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2025

Keywords

Crossrefs

Cf. A006153, A380017, A380019.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (-2)^k * k^(n-k) * binomial(-1/2,k)/(n-k)!.
a(n) ~ sqrt(2) * n^n / (sqrt(1 + LambertW(1/2)) * exp(n) * LambertW(1/2)^n). - Vaclav Kotesovec, Jan 23 2025

A385311 Expansion of e.g.f. 1/(1 - 3 * x * cos(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 25, 232, 2805, 41920, 744933, 15340416, 359136073, 9419223040, 273558859409, 8714789788672, 302151400126589, 11326084055150592, 456421403198919325, 19677025400034590720, 903660903945306053137, 44042354270955276599296, 2270411632567521580120713
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A352252, A385310.
Cf. A380017, A385305, A385307, A385309.
Cf. A007559, A185951, A352647, A385284.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*I^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A007559(k) * i^(n-k) * A185951(n,k), where i is the imaginary unit and A185951(n,0) = 0^n.

A380039 E.g.f. A(x) satisfies A(x) = 1/( 1 - 3*x*exp(x*A(x)) )^(1/3).

Original entry on oeis.org

1, 1, 6, 61, 908, 17865, 438286, 12901735, 443475432, 17443879057, 773018191610, 38117147134671, 2070381313048588, 122841147634754185, 7905667340470592070, 548555101319868261655, 40825552788531622527056, 3244188226183716688784289, 274164589130871765969460594
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2025

Keywords

Crossrefs

Cf. A380017, A380041.
Cf. A380040.

Programs

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

Formula

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

A385305 Expansion of e.g.f. 1/(1 - 3 * sinh(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 29, 296, 3921, 63904, 1236509, 27700096, 705098241, 20100847104, 634406699389, 21959759364096, 827184049670161, 33684401687855104, 1474548883501060669, 69051807696652599296, 3444499079760040247681, 182339939994632235515904, 10209271857672376613472349
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A006154, A385304.
Cf. A380017, A385307, A385309, A385311.
Cf. A007559, A136630, A235135.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} A007559(k) * A136630(n,k).
a(n) ~ sqrt(Pi) * 2^(1/3) * n^(n - 1/6) / (5^(1/6) * Gamma(1/3) * exp(n) * log((1 + sqrt(10))/3)^(n + 1/3)). - Vaclav Kotesovec, Jun 28 2025

A385307 Expansion of e.g.f. 1/(1 - 3 * sin(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 27, 264, 3361, 52704, 981707, 21176704, 519150241, 14255163904, 433384277787, 14451212550144, 524406240059521, 20572970822959104, 867641565719168267, 39145118179183427584, 1881294510800399083201, 95950279080398196834304, 5176039012712211526485147
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A000111, A385306.
Cf. A380017, A385305, A385309, A385311.
Cf. A007559, A007788, A136630.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*I^(n-k)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} A007559(k) * i^(n-k) * A136630(n,k), where i is the imaginary unit.
a(n) ~ n! / (sqrt(2) * Gamma(1/3) * n^(2/3) * arcsin(1/3)^(n + 1/3)). - Vaclav Kotesovec, Jun 28 2025

A385309 Expansion of e.g.f. 1/(1 - 3 * x * cosh(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 31, 328, 4485, 75520, 1509347, 34916224, 917703145, 27011107840, 880133628231, 31451749424128, 1223047891889837, 51414400611438592, 2323391075748100555, 112315439676217262080, 5783449255108473820497, 316034972288791445241856, 18265740423344520141491951
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A205571, A385308.
Cf. A380017, A385305, A385307, A385311.
Cf. A007559, A185951, A352649, A385282.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A007559(k) * A185951(n,k), where A185951(n,0) = 0^n.

A380019 Expansion of e.g.f. 1/(1 - 4*x*exp(x))^(1/4).

Original entry on oeis.org

1, 1, 7, 78, 1249, 26100, 673101, 20655082, 735030913, 29759100264, 1350726180085, 67929497104326, 3749296817347137, 225321905599163308, 14646040616615433949, 1023818460912628352490, 76589660469522857865601, 6105092923000191785913552, 516586509938516858800548453
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2025

Keywords

Crossrefs

Cf. A006153, A380015, A380017.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (-4)^k * k^(n-k) * binomial(-1/4,k)/(n-k)!.

A380041 E.g.f. A(x) satisfies A(x) = 1/( 1 - 3*x*exp(x*A(x)^2) )^(1/3).

Original entry on oeis.org

1, 1, 6, 67, 1124, 25325, 718606, 24629395, 990296504, 45718478137, 2383877762810, 138578689119431, 8887132981365508, 623319005140469989, 47465740413056117894, 3900149351529967753435, 343951717449176947732976, 32405206661688405897284849, 3248370338004030319683766642
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2025

Keywords

Crossrefs

Cf. A380017, A380039.

Programs

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

Formula

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

A380134 Expansion of e.g.f. (1 + 3*x*exp(x))^(1/3).

Original entry on oeis.org

1, 1, 0, 1, -4, 25, -194, 1813, -19816, 248113, -3502630, 55052701, -953576876, 18048491305, -370623627178, 8207063150245, -194950421191504, 4944881412682081, -133394451535683278, 3813510163227155245, -115170227064335439700, 3663942710200202043481
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2025

Keywords

Crossrefs

Cf. A028310, A380133.
Cf. A380051, A380094.
Cf. A380017.

Programs

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

Formula

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

A380029 Expansion of e.g.f. (1 - 3*x*exp(x))^(1/3).

Original entry on oeis.org

1, -1, -4, -25, -252, -3545, -63806, -1397781, -36069272, -1071165745, -35977484250, -1348257912221, -55766033179220, -2523251585908521, -123972318738063446, -6572554273909419685, -373979858167243433136, -22731929051273411113313, -1470009560015441800798514
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2025

Keywords

Crossrefs

Cf. A004990, A380017, A380030.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (-3)^k * k^(n-k) * binomial(1/3,k)/(n-k)!.
Showing 1-10 of 12 results. Next

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