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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A381273 Expansion of e.g.f. exp(x * cosh(2*x)).

Original entry on oeis.org

1, 1, 1, 13, 49, 201, 2161, 12629, 102817, 1118161, 9109921, 105660765, 1223720785, 13461561881, 186666204817, 2406325357861, 33607592404033, 516511765519521, 7658010172957249, 126206019752173997, 2115466479287184241, 36218229615683409001, 666810643855970901937
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. A003727, A381274.
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^(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^(n-k) * A185951(n,k).

A381274 Expansion of e.g.f. exp(x * cosh(3*x)).

Original entry on oeis.org

1, 1, 1, 28, 109, 676, 10261, 65584, 881497, 11930896, 122708521, 2186539840, 30542901445, 477545743936, 9168255077437, 149358238356736, 3043023842477233, 61000460650291456, 1225825910880514129, 28395625697194028032, 621110654837608378141, 14936817377079335166976
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. A003727, A381273.
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, 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-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} 3^(n-k) * A185951(n,k).

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.

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).

A381282 Expansion of e.g.f. 1/(1 - x * cos(2*x)).

Original entry on oeis.org

1, 1, 2, -6, -72, -520, -1200, 24752, 516992, 5106816, 5287680, -998945024, -23719719936, -272471972864, 1326261594112, 149170761246720, 3843177252618240, 42752553478356992, -863092250325614592, -59317347865870139392, -1577115871098630307840, -13173264127625587851264
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. A352252, A381283.
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*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,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * (2*i)^(n-k) * A185951(n,k), where i is the imaginary unit.

A381447 E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^2 * cosh(x*A(x)^2).

Original entry on oeis.org

1, 1, 4, 33, 432, 7745, 175680, 4818457, 155138816, 5738752161, 239890406400, 11184338164241, 575437530083328, 32387311520034913, 1979498673768132608, 130566701113312750665, 9244392468538216611840, 699309477932976288024257, 56289911059840766752456704
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Comments

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

Links

Crossrefs

Cf. A381171, A381448.
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(2*n+1, k)*a185951(n, k))/(2*n+1);

Formula

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

A385283 Expansion of e.g.f. 1/(1 - 2 * x * cos(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 3, -39, -775, -9045, -85813, -426447, 7321329, 325555155, 7786757011, 137053423881, 1388713844713, -21121997539461, -1827406866674085, -69034283067822495, -1852635543265039903, -30574875232261547613, 308376017794648053539, 54871741689019890859065
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A352252, A385284.
Cf. A001586, A235134, A380155, A385281.
Cf. A001147, A185951.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*(2*I)^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A001147(k) * (2*i)^(n-k) * A185951(n,k), where i is the imaginary unit and A185951(n,0) = 0^n.

A381140 Expansion of e.g.f. exp( -LambertW(-x * cosh(x)) ).

Original entry on oeis.org

1, 1, 3, 19, 161, 1781, 24667, 409991, 7959233, 176920489, 4432942931, 123648692795, 3800647961761, 127654261471517, 4651982506605995, 182824074836850991, 7708128977570816129, 347059689259637711441, 16621016953663100702755, 843658152872351669816675
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. A003727, A162649, A381143.
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)*a185951(n, k));

Formula

E.g.f. A(x) satisfies A(x) = exp( x * cosh(x) * A(x) ).
a(n) = Sum_{k=0..n} (k+1)^(k-1) * A185951(n,k).
Previous Showing 31-40 of 77 results. Next

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