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 11-20 of 20 results.

A352649 Expansion of e.g.f. 1/(1 - 3 * x * cosh(x)).

Original entry on oeis.org

1, 3, 18, 171, 2160, 34035, 643680, 14203371, 358178688, 10161542691, 320315005440, 11106766229163, 420132741912576, 17216605635562515, 759789379494512640, 35925442734363182955, 1811923104577065615360, 97097117111612660889411, 5509300889675218610552832
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Crossrefs

Cf. A205571, A352648.
Cf. A185951.

Programs

  • Mathematica
    With[{m = 17}, Range[0, m]! * CoefficientList[Series[1/(1 - 3*x*Cosh[x]), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*x*cosh(x))))
  • PARI
    a(n) = if(n==0, 1, 3*sum(k=0, (n-1)\2, (2*k+1)*binomial(n, 2*k+1)*a(n-2*k-1)));

Formula

a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} 3^k * k! * A185951(n,k). - Seiichi Manyama, Jun 25 2025

A381206 Expansion of e.g.f. 1/(1 - x*cosh(x))^2.

Original entry on oeis.org

1, 2, 6, 30, 192, 1450, 12960, 133574, 1550976, 20055186, 285903360, 4452231982, 75186726912, 1368588922298, 26709799753728, 556339845854550, 12318065768693760, 288894650033594914, 7154212267816648704, 186545064693433665854, 5108590743587243950080
Offset: 0

Views

Author

Seiichi Manyama, Feb 17 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A205571, A381207.
Cf. A185951.

Programs

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

Formula

a(n) = Sum_{k=0..n} (k+1)! * A185951(n,k).

A381281 Expansion of e.g.f. 1/(1 - x * cosh(3*x)).

Original entry on oeis.org

1, 1, 2, 33, 240, 2145, 33120, 480753, 7878528, 158696577, 3384322560, 78934776129, 2053186983936, 57231998680545, 1714372871178240, 55323775198258065, 1899762412262031360, 69264871449203672577, 2677542944055160209408, 109197154520146527569505
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A205571, A381280.
Cf. A185951.

Programs

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

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 9^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * 3^(n-k) * A185951(n,k).

A385282 Expansion of e.g.f. 1/(1 - 3 * x * cosh(3*x))^(1/3).

Original entry on oeis.org

1, 1, 4, 55, 712, 11605, 248320, 6218443, 178519936, 5846857993, 214490045440, 8700546508159, 387053184719872, 18737207168958109, 980424546959183872, 55142056940797803475, 3317502712746788945920, 212592531182720568805777, 14456626429227650204041216
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A205571, A385281.
Cf. A007559, A185951.
Cf. A069814.

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)*3^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A007559(k) * 3^(n-k) * A185951(n,k), where A185951(n,0) = 0^n.
a(n) ~ sqrt(2*Pi) * 3^n * n^(n - 1/6) / (Gamma(1/3) * (1/r + sqrt(1 - r^2))^(1/3) * exp(n) * r^(n + 1/3)), where r = A069814. - Vaclav Kotesovec, Jun 24 2025

A352250 Expansion of e.g.f. 1 / (1 - x * sin(x)) (even powers only).

Original entry on oeis.org

1, 2, 20, 486, 21944, 1591210, 169207092, 24808395262, 4796420822384, 1182349445882706, 361939981107422060, 134705596642758848806, 59900689507397744253096, 31365504832631796986962426, 19102102945852191813235300004, 13387748268024668296590660222030
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

Crossrefs

Cf. A000111, A000364, A009214, A205571, A210657, A302397, A352251, A352252.

Programs

  • Mathematica
    nmax = 30; Take[CoefficientList[Series[1/(1 - x Sin[x]), {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = 2 Sum[(-1)^(k + 1) Binomial[2 n, 2 k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]
  • PARI
    my(x='x+O('x^40), v=Vec(serlaplace(1 /(1-x*sin(x))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

Formula

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

A352253 Expansion of e.g.f. 1 / (1 - x * sinh(x) / 2) (even powers only).

Original entry on oeis.org

1, 1, 8, 153, 5492, 316625, 26774622, 3121729709, 479962730648, 94087054172673, 22904161764512570, 6778870099212235805, 2397161662661680925364, 998186321121004312238513, 483430830256916593106991782, 269435322393253822641626419725, 171224984800186115316322226731952
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

Crossrefs

Cf. A006153, A006154, A094088, A205571, A352251, A352254.

Programs

  • Mathematica
    nmax = 32; Take[CoefficientList[Series[1/(1 - x Sinh[x]/2), {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n, 2 k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]
  • PARI
    my(x='x+O('x^40), v=Vec(serlaplace(1 /(1-x*sinh(x)/2)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n,2*k) * k * a(n-k).

A381344 Expansion of e.g.f. 1/( 1 - x * cosh(sqrt(2)*x) ).

Original entry on oeis.org

1, 1, 2, 12, 72, 500, 4560, 47936, 565376, 7572240, 112838400, 1844425792, 32910332928, 636463467328, 13251265570816, 295598326909440, 7034150340034560, 177843592245969152, 4760839037033054208, 134528586280018721792, 4001489050575059025920, 124973219149863342633984
Offset: 0

Views

Author

Seiichi Manyama, Feb 20 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A205571, A352252, A381280, A381281, A381282, A381283, A381345.
Cf. A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*2^((n-k)/2)*a185951(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 2^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * 2^((n-k)/2) * A185951(n,k).
a(n) ~ sqrt(Pi) * 2^(n/2 + 1) * n^(n + 1/2) / ((cosh(r) + r*sinh(r)) * exp(n) * r^(n+1)), where r = 0.95090803593755778120914299086438615849657408871... is the root of the equation r*cosh(r) = sqrt(2). - Vaclav Kotesovec, Apr 19 2025

A381345 Expansion of e.g.f. 1/( 1 - x * cos(sqrt(2)*x) ).

Original entry on oeis.org

1, 1, 2, 0, -24, -220, -1200, -2576, 52864, 1016208, 10909440, 57039488, -687971328, -26190716864, -450123634688, -4238375059200, 24514848522240, 2156422420074752, 54984136073084928, 799573460292407296, 42320889956270080, -425007017470737816576, -15563879892284330213376
Offset: 0

Views

Author

Seiichi Manyama, Feb 20 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A205571, A352252, A381280, A381281, A381282, A381283, A381344.
Cf. A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*(-2)^((n-k)/2)*a185951(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-2)^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * (-2)^((n-k)/2) * A185951(n,k).

A381445 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * cosh(x) * A(x)^2 ).

Original entry on oeis.org

1, 1, 6, 75, 1392, 34925, 1108080, 42562807, 1920796416, 99628495353, 5840628226560, 381927689957891, 27562916396961792, 2176123474607538469, 186580455503952427008, 17264834430223073672175, 1714909152672462179205120, 182002038900785304200753777, 20553746198157175799599202304
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A205571, A295256, A381446.
Cf. A185951.

Programs

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

Formula

a(n) = Sum_{k=0..n} k! * binomial(3*k+1,k)/(3*k+1) * A185951(n,k).

A381446 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * cosh(x) * A(x)^3 ).

Original entry on oeis.org

1, 1, 8, 135, 3456, 120245, 5303040, 283559227, 17830210048, 1289406976713, 105435719470080, 9619902621234191, 968905466782150656, 106779534666615500989, 12781543241568143171584, 1651368425166943566943875, 229049483642619517308764160, 33947359023461155854768564497
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A205571, A295256, A381445.
Cf. A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(4*k+1, k)/(4*k+1)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(4*k+1,k)/(4*k+1) * A185951(n,k).
Previous Showing 11-20 of 20 results.

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