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.

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A381284 Expansion of e.g.f. 1/(1 - sinh(3*x) / 3).

Original entry on oeis.org

1, 1, 2, 15, 96, 741, 7632, 87795, 1149696, 17155881, 282880512, 5128464375, 101592631296, 2178698451021, 50314379323392, 1245198047833755, 32868161979088896, 921803465256094161, 27373850876851126272, 858044392807801699935, 28311289100161039466496
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2025

Keywords

Crossrefs

Cf. A006154, A191277.
Cf. A136630.

Programs

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

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 9^k * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * 3^(n-k) * A136630(n,k).
a(n) ~ sqrt(Pi/5) * 3^(n+1) * n^(n + 1/2) / (arcsinh(3)^(n+1) * exp(n)). - Vaclav Kotesovec, Apr 19 2025

A381386 E.g.f. A(x) satisfies A(x) = 1/( 1 - sinh(x * A(x)^2) ).

Original entry on oeis.org

1, 1, 6, 73, 1360, 34321, 1095584, 42350673, 1923628032, 100430070721, 5926517800192, 390116250605401, 28341322114027520, 2252512575040254801, 194421212092585943040, 18110799663166635386017, 1810994441189833169698816, 193488658627430346315888385, 21997611392941496027173879808
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2025

Keywords

Crossrefs

Cf. A136630, A381387.

Programs

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

Formula

a(n) = Sum_{k=0..n} k! * binomial(2*n+k+1,k)/(2*n+k+1) * A136630(n,k).
E.g.f.: ( (1/x) * Series_Reversion( x*(1 - sinh(x))^2 ) )^(1/2).

A381387 E.g.f. A(x) satisfies A(x) = 1/( 1 - sinh(x * A(x)) )^2.

Original entry on oeis.org

1, 2, 14, 182, 3520, 91002, 2954400, 115638014, 5303063552, 278979672050, 16565016146176, 1095997724407302, 79966475806040064, 6379010456725968362, 552344502268240535552, 51595059327775839277646, 5171865567269556457308160, 553764742712510134123863522
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2025

Keywords

Crossrefs

Cf. A136630, A381386.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381386.
a(n) = 2 * Sum_{k=0..n} k! * binomial(2*n+k+2,k)/(2*n+k+2) * A136630(n,k).
E.g.f.: (1/x) * Series_Reversion( x*(1 - sinh(x))^2 ).

A381388 E.g.f. A(x) satisfies A(x) = 1/( 1 - sin(x * A(x)^2) ).

Original entry on oeis.org

1, 1, 6, 71, 1280, 31201, 961184, 35838991, 1569696768, 79007365921, 4494170889472, 285130996517399, 19963494971809792, 1529055924661457921, 127179971644212387840, 11416028319985437309215, 1099976414821996358795264, 113239907265894992879189185, 12404749306625020735299780608
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2025

Keywords

Crossrefs

Cf. A136630, A381389.

Programs

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

Formula

a(n) = Sum_{k=0..n} k! * binomial(2*n+k+1,k)/(2*n+k+1) * i^(n-k) * A136630(n,k), where i is the imaginary unit.
E.g.f.: ( (1/x) * Series_Reversion( x*(1 - sin(x))^2 ) )^(1/2).

A381389 E.g.f. A(x) satisfies A(x) = 1/( 1 - sin(x * A(x)) )^2.

Original entry on oeis.org

1, 2, 14, 178, 3344, 83722, 2628000, 99358810, 4398573568, 223280915090, 12788876882176, 816044058415298, 57411735641690112, 4415467258014111002, 368568207039291072512, 33186631279383615035242, 3206409506796711229521920, 330893672854541429428877602
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2025

Keywords

Crossrefs

Cf. A136630, A381388.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381388.
a(n) = 2 * Sum_{k=0..n} k! * binomial(2*n+k+2,k)/(2*n+k+2) * i^(n-k) * A136630(n,k), where i is the imaginary unit.
E.g.f.: (1/x) * Series_Reversion( x*(1 - sin(x))^2 ).

A381415 E.g.f. A(x) satisfies A(x) = exp( sinh(x * A(x)^2) ).

Original entry on oeis.org

1, 1, 5, 50, 765, 15852, 415441, 13182976, 491502521, 21061603152, 1020066862269, 55107133707232, 3285531022228725, 214295961023511616, 15179005200468020489, 1160334809344169734144, 95214513195493336071537, 8347897781857074205573376, 778804910740650550851809013
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Crossrefs

Cf. A136630, A162650.

Programs

  • Mathematica
    Join[{1}, Table[Sum[(2*n + 1)^(k-1) / (2^k*k!) * Sum[(-1)^(k-j) * (2*j - k)^n * Binomial[k, j], {j, 0, k}], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jul 04 2025 *)
  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, (2*n+1)^(k-1)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} (2*n+1)^(k-1) * A136630(n,k).
a(n) ~ s * n^(n-1) / (2*sqrt(1 + r*s^2*sqrt(1 - 4*r^2*s^4)) * exp(n) * r^n), where r = 0.1774317812751606880070098054556619184142424898705... and s = 1.597465072615091018021826608474818660705268320323... are the roots of the system of equations exp(sinh(r*s^2)) = s, 2*r*s^2*cosh(r*s^2) = 1. - Vaclav Kotesovec, Jul 04 2025

A381442 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + sinh(x))^2 ).

Original entry on oeis.org

1, 2, 10, 86, 1080, 18042, 377936, 9538622, 281946496, 9557102450, 365548361472, 15576454300134, 731807446707200, 37584596599753322, 2094995668172597248, 125966553940498047182, 8127048592610380578816, 560040497770823162810082, 41054563701320694564061184
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Links

Crossrefs

Cf. A136630, A198865.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = (1 + sinh(x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A198865.
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(2*n+2,k) * A136630(n,k).

A381443 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + sinh(x))^3 ).

Original entry on oeis.org

1, 3, 24, 333, 6720, 179523, 5992800, 240498261, 11287790592, 607019415075, 36813049552896, 2486167829854173, 185070328813031424, 15056826823777670883, 1329283990371617820672, 126573877370649849898149, 12930948581449447912243200, 1410875453109072905123881923
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Links

Crossrefs

Cf. A377554, A381450.
Cf. A136630, A381430.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = (1 + sinh(x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A381430.
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(3*n+3,k) * A136630(n,k).

A381519 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + sin(x))^2 ).

Original entry on oeis.org

1, 2, 10, 82, 936, 13642, 240656, 4952218, 115608704, 2992207250, 84070140672, 2507383885730, 77117178496000, 2329071118971482, 61202811821836288, 690380688651775978, -88097620429234470912, -11900508444760552311518, -1112180862634722333884416
Offset: 0

Views

Author

Seiichi Manyama, Feb 26 2025

Keywords

Links

Crossrefs

Cf. A136630, A381389, A381518.
Cf. A377553, A381442, A381449, A381521.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = (1 + sin(x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381518.
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(2*n+2,k) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

A352639 Expansion of e.g.f. exp(2*sin(x)).

Original entry on oeis.org

1, 2, 4, 6, 0, -46, -192, -266, 1792, 14114, 34816, -171930, -2027520, -6522382, 34750464, 496296022, 1748500480, -12731696062, -186550845440, -617309234490, 7292215885824, 99199654760978, 248883934396416, -5836506132182090, -69729013345550336
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Crossrefs

Cf. A002017, A136630, A352640.
Cf. A007289, A352279, A352642.

Programs

  • Mathematica
    With[{m = 24}, Range[0, m]! * CoefficientList[Series[Exp[2*Sin[x]], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*sin(x))))
  • PARI
    a(n) = if(n==0, 1, 2*sum(k=0, (n-1)\2, (-1)^k*binomial(n-1, 2*k)*a(n-2*k-1)));

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} 2^k * i^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 18 2025
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