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 14 results. Next

A352252 Expansion of e.g.f. 1 / (1 - x * cos(x)).

Original entry on oeis.org

1, 1, 2, 3, 0, -55, -480, -3157, -15232, -16623, 898560, 16316179, 194574336, 1666248025, 5418649600, -170157839685, -5164467978240, -92955464490463, -1188910801354752, -7329026447550685, 157257042777866240, 7516793832172469481, 187200588993188069376
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

Links

Crossrefs

Cf. A000111, A007289, A009189, A094088, A205571, A302397, A352250, A352251.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 - x Cos[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[(-1)^k Binomial[n, 2 k + 1] (2 k + 1) a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 22}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(1 / (1 - x * cos(x)))) \\ Michel Marcus, Mar 10 2022
  • 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!*I^(n-k)*a185951(n, k)); \\ Seiichi Manyama, Feb 17 2025

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n,2*k+1) * (2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 17 2025

A352642 Expansion of e.g.f. exp(2 * x * cos(x)).

Original entry on oeis.org

1, 2, 4, 2, -32, -198, -416, 2634, 30720, 107378, -605696, -10282094, -46020608, 304968874, 6121832448, 29994597338, -279697555456, -5729595393310, -26849178681344, 401845799334690, 7714801999937536, 29062583111892506, -812705956979802112
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Links

Crossrefs

Cf. A009189, A185951, A352643.

Programs

  • Mathematica
    With[{m = 22}, Range[0, m]! * CoefficientList[Series[Exp[2*x*Cos[x]], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*x*cos(x))))
  • PARI
    a(n) = if(n==0, 1, 2*sum(k=0, (n-1)\2, (-1)^k*(2*k+1)*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 * (2*k+1) * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} 2^k * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 18 2025

A352643 Expansion of e.g.f. exp(3 * x * cos(x)).

Original entry on oeis.org

1, 3, 9, 18, -27, -552, -3051, -3504, 102825, 1043712, 3192129, -41548416, -653192883, -3033406464, 31367500173, 670266381312, 3916411302609, -40460110970880, -1038593550985479, -6810646726410240, 82445831323038261, 2280185182260854784, 15300402721484153733
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2022

Keywords

Links

Crossrefs

Cf. A009189, A185951, A352642.

Programs

  • Mathematica
    With[{m = 22}, Range[0, m]! * CoefficientList[Series[Exp[3*x*Cos[x]], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(3*x*cos(x))))
  • PARI
    a(n) = if(n==0, 1, 3*sum(k=0, (n-1)\2, (-1)^k*(2*k+1)*binomial(n-1, 2*k)*a(n-2*k-1)));

Formula

a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} (-1)^k * (2*k+1) * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} 3^k * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 18 2025

A349103 Expansion of e.g.f. exp( (sin(x) + x*cos(x))/2 ).

Original entry on oeis.org

1, 1, 1, -1, -7, -16, 19, 270, 809, -1832, -26939, -81680, 415817, 5085192, 11943199, -154023064, -1510744111, -1192784896, 80798373289, 615743987520, -1348945040647, -55570193758592, -296824726927253, 2567750854163200, 47111874571146041
Offset: 0

Views

Author

Seiichi Manyama, Mar 26 2022

Keywords

Links

Crossrefs

Cf. A002017, A009189, A349104, A351937.

Programs

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

Formula

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

A381141 Expansion of e.g.f. exp( -LambertW(-x * cos(x)) ).

Original entry on oeis.org

1, 1, 3, 13, 89, 821, 9667, 137817, 2306705, 44308009, 960645251, 23205700453, 618086944873, 17996847978461, 568729575572355, 19387150575025201, 709130794848586657, 27704208465508996945, 1151379111946617111043, 50721472225191792506301, 2360928161776701549045241
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 2025

Keywords

Comments

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

Links

Crossrefs

Cf. A009189, A381144, A381146.
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)^(k-1)*I^(n-k)*a185951(n, k));

Formula

E.g.f. A(x) satisfies A(x) = exp( x * cos(x) * A(x) ).
a(n) = Sum_{k=0..n} (k+1)^(k-1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.

A381275 Expansion of e.g.f. exp(x * cos(2*x)).

Original entry on oeis.org

1, 1, 1, -11, -47, -39, 1681, 10893, -13215, -851471, -5515679, 34375397, 887687857, 3982645577, -85350572943, -1466457337859, -659043831871, 270733024430305, 3181606182917569, -24432689736388395, -1076204061663657839, -6834631528147762247, 221729710998069153617
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. A009189, A381276.
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, (2*I)^(n-k)*a185951(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-4)^k * (2*k+1) * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} (2*i)^(n-k) * A185951(n,k), where i is the imaginary unit.

A381276 Expansion of e.g.f. exp(x * cos(3*x)).

Original entry on oeis.org

1, 1, 1, -26, -107, 136, 9181, 53488, -427895, -10486016, -43859879, 1373548672, 23512856797, -30564574208, -6412871847563, -73709639926784, 1060067525174929, 40587133606543360, 179320588932698929, -14474677657838059520, -306563699887974043739, 2301792469199499132928
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. A009189, A381275.
Cf. A185951, A352643, A381283.

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, (3*I)^(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-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} (3*i)^(n-k) * A185951(n,k), where i is the imaginary unit.

A354517 Expansion of e.g.f. cos(x)^exp(x).

Original entry on oeis.org

1, 0, -1, -3, -5, 10, 134, 742, 2325, -2820, -118756, -1138368, -7132025, -20945990, 196411214, 4438271692, 50498101545, 400644382200, 1571151012344, -16415635331328, -500300343321365, -7486919544207050, -81415563206142166, -563533196469890228
Offset: 0

Views

Author

Seiichi Manyama, Aug 16 2022

Keywords

Crossrefs

Cf. A000248, A009189, A215515, A354518, A354519.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(cos(x)^exp(x)))
  • PARI
    a354519(n) = sum(k=1, n\2, ((-4)^k-(-16)^k)*bernfrac(2*k)/(2*k)*binomial(n, 2*k));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, a354519(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} A354519(k) * binomial(n-1,k-1) * a(n-k).

A381144 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * cos(x)) ).

Original entry on oeis.org

1, 1, 3, 13, 65, 221, -2933, -120903, -3104127, -71637191, -1562635789, -31373685947, -505087300991, -1692007785259, 402032879446395, 28152810613025521, 1423083552938781697, 62552808878706976625, 2459148829654813484131, 82692880516086149155581
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 2025

Keywords

Comments

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

Links

Crossrefs

Cf. A009189, A381141, A381146.
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, (n+1)^(k-1)*I^(n-k)*a185951(n, k));

Formula

E.g.f. A(x) satisfies A(x) = exp( x * A(x) * cos(x * A(x)) ).
a(n) = Sum_{k=0..n} (n+1)^(k-1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.

A381146 E.g.f. A(x) satisfies A(x) = exp( x * cos(x * A(x)) ).

Original entry on oeis.org

1, 1, 1, -2, -35, -264, -1019, 9864, 302905, 4181824, 23080201, -632195200, -25239729899, -478790195584, -2682065360883, 188875659540736, 8843706554450033, 203538869514047488, 751681101659548177, -169782541027003551744, -8866196526809624969139
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 2025

Keywords

Comments

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

Crossrefs

Cf. A009189, A381141, A381144.
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, (n-k+1)^(k-1)*I^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * i^(n-k) * A185951(n,k), where i is the imaginary unit.
Showing 1-10 of 14 results. Next

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