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 77 results. Next

A385343 Exponential Riordan array (1, arcsin(x)).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 4, 0, 1, 0, 9, 0, 10, 0, 1, 0, 0, 64, 0, 20, 0, 1, 0, 225, 0, 259, 0, 35, 0, 1, 0, 0, 2304, 0, 784, 0, 56, 0, 1, 0, 11025, 0, 12916, 0, 1974, 0, 84, 0, 1, 0, 0, 147456, 0, 52480, 0, 4368, 0, 120, 0, 1, 0, 893025, 0, 1057221, 0, 172810, 0, 8778, 0, 165, 0, 1
Offset: 0

Views

Author

Seiichi Manyama, Jun 26 2025

Keywords

Examples

			Triangle starts:
  1;
  0,   1;
  0,   0,  1;
  0,   1,  0,   1;
  0,   0,  4,   0,  1;
  0,   9,  0,  10,  0,  1;
  0,   0, 64,   0, 20,  0, 1;
  0, 225,  0, 259,  0, 35, 0, 1;

Crossrefs

Essentialy same as A121408.
Row sums give A006228.
Cf. A091885, A136630, A185951.

Programs

  • PARI
    T(n, k) = my(x='x+O('x^(n+1))); n!*polcoef(asin(x)^k/k!, n);

Formula

E.g.f. of column k (with leading zeros): arcsin(x)^k / k!
T(n,k) = A121408(n,k) for k > 0.

A205571 Expansion of e.g.f. 1/(1 - x*cosh(x)).

Original entry on oeis.org

1, 1, 2, 9, 48, 305, 2400, 22057, 230272, 2708001, 35412480, 509177801, 7986468864, 135718942801, 2483729876992, 48699677975145, 1018542257111040, 22634000289407297, 532557637644976128, 13226748101381102473, 345792863300174479360, 9492229607399841038961
Offset: 0

Views

Author

Paul D. Hanna, Jan 28 2012

Keywords

Comments

Radius of convergence of e.g.f. is |x| < r where r = 0.7650099545507... satisfies cosh(r) = 1/r. See A069814.

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 48*x^4/4! + 305*x^5/5! +...

Links

Crossrefs

Cf. A069814, A185951.
Cf. A000111, A006154, A352252.
Cf. A381206, A381207.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-x*Cosh[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 13 2013 *)
  • PARI
    {a(n)=n!*polcoeff(1/(1-x*cosh(x +x*O(x^n))),n)}
  • 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!*a185951(n, k)); \\ Seiichi Manyama, Feb 17 2025

Formula

a(2*n-1) == 1 (mod 4), a(2*n+2) == 0 (mod 4), for n>=1.
a(n) ~ n!/(1+r*sqrt(1-r^2))*(1/r)^n, where r = A069814 = 0.7650099545507321... is the root of the equation r*cosh(r)=1. - Vaclav Kotesovec, Feb 13 2013
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * (2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 10 2022
a(n) = Sum_{k=0..n} k! * A185951(n,k). - Seiichi Manyama, Feb 17 2025

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

A381171 Expansion of e.g.f. (1/x) * Series_Reversion( x / (1 + x*cosh(x)) ).

Original entry on oeis.org

1, 1, 2, 9, 72, 725, 8640, 124117, 2117248, 41477193, 913305600, 22371549761, 604476094464, 17858943664861, 572524035586048, 19793963392789965, 734249332747960320, 29090332675789113617, 1225991945551031304192, 54765451909152748484857, 2584803582762012599910400
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2025

Keywords

Comments

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

Links

Crossrefs

Cf. A162654, A215364, A381172.
Cf. A162653, A381173, A381181.
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(n+1, k)*a185951(n, k))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = 1/( 1 - x * cosh(x * A(x)) ).
a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(n+1,k) * A185951(n,k).

A295256 Expansion of e.g.f. 2/(1 + sqrt(1 - 4*x*cosh(x))).

Original entry on oeis.org

1, 1, 4, 33, 384, 5945, 115680, 2713417, 74568704, 2350925649, 83660474880, 3317599815761, 145087264278528, 6937450761100873, 360078818344534016, 20162761727269502265, 1211588127198611374080, 77769423447774393465377, 5310706204624302598127616, 384439720034220718046773249
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 18 2017

Keywords

Crossrefs

Cf. A000108, A185951, A295237, A295254, A295255, A295257, A295258.

Programs

  • Maple
    a:=series(2/(1+sqrt(1-4*x*cosh(x))),x=0,21): seq(n!*coeff(a,x,n),n=0..19); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[2/(1 + Sqrt[1 - 4 x Cosh[x]]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-x Cosh[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1/(1 - x*cosh(x)/(1 - x*cosh(x)/(1 - x*cosh(x)/(1 - x*cosh(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(2 + 2*r*sqrt(1-16*r^2)) * n^(n-1) / (exp(n) * r^n), where r = 0.2428073624074744554637516823... is the root of the equation 2*r*(exp(2*r)+1) = exp(r). - Vaclav Kotesovec, Nov 18 2017
a(n) = Sum_{k=0..n} k! * binomial(2*k+1,k)/(2*k+1) * A185951(n,k). - Seiichi Manyama, Feb 23 2025

A381173 Expansion of e.g.f. (1/x) * Series_Reversion( x / (1 + x*cos(x)) ).

Original entry on oeis.org

1, 1, 2, 3, -24, -475, -5760, -52297, -155008, 8781705, 313344000, 6966991339, 102864807936, 18664712365, -71473582229504, -3387816787568865, -103478592573112320, -1899945146589964783, 18941335827815596032, 3808766537454425974739, 215681241589289359769600
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2025

Keywords

Comments

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

Links

Crossrefs

Cf. A381174, A381175, A381176.
Cf. A162653, A381171, A381181.
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(n+1, k)*I^(n-k)*a185951(n, k))/(n+1);

Formula

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

A381207 Expansion of e.g.f. 1/(1 - x*cosh(x))^3.

Original entry on oeis.org

1, 3, 12, 69, 504, 4335, 43200, 490161, 6220032, 87242427, 1340305920, 22375475133, 403237638144, 7801208775399, 161245892161536, 3545854432602345, 82653484859228160, 2035605515838402291, 52814589875313573888, 1439814136866851346357, 41145786213980645621760
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. A377530, A381209, A381210, A381211.
Cf. A205571, A381206.
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)!*a185951(n, k))/2;

Formula

a(n) = 1/2 * Sum_{k=0..n} (k+2)! * A185951(n,k).

A381376 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * cosh(x * A(x)^2) ).

Original entry on oeis.org

1, 1, 2, 9, 96, 1385, 22080, 403417, 8829184, 227956689, 6667822080, 215780258441, 7674505073664, 298885308910201, 12661212551163904, 578940699178779225, 28400662193828659200, 1488075298726340008097, 82965096417136263561216, 4904558063539270185865609
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2025

Keywords

Comments

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

Crossrefs

Cf. A381171, A381300.
Cf. A185951, A381377.

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-k+1, k)/(2*n-k+1)*a185951(n, k));

Formula

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

A381378 E.g.f. A(x) satisfies A(x) = 1/( 1 - x * cos(x * A(x)^2) ).

Original entry on oeis.org

1, 1, 2, 3, -48, -1135, -18240, -231637, -1356544, 53849889, 3026119680, 100808786419, 2429052865536, 26284690243825, -1539261873164288, -140633348417624805, -7196339681250508800, -258335768147494234303, -4225401456668904259584, 307227604973975435785571
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2025

Keywords

Comments

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

Crossrefs

Cf. A364980, A381376, A381382, A381384.
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-k+1, k)/(2*n-k+1)*I^(n-k)*a185951(n, k));

Formula

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

A385310 Expansion of e.g.f. 1/(1 - 2 * x * cos(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 12, 69, 500, 4455, 46928, 571977, 7914384, 122585355, 2100940864, 39470867469, 806555184448, 17808628411119, 422498774818560, 10717948285126545, 289501146405400832, 8295124400250875667, 251300745071590317056, 8025654235707259740885, 269482309052945201181696
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A352252, A385311.
Cf. A380015, A385304, A385306, A385308.
Cf. A001147, A185951, A352646, A385283.

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

Formula

a(n) = Sum_{k=0..n} A001147(k) * i^(n-k) * A185951(n,k), where i is the imaginary unit and A185951(n,0) = 0^n.
Showing 1-10 of 77 results. Next

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