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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A219503 Expansion of e.g.f. Sum_{n>=0} (n+1)^(n-1) * sinh(x)^n / n!.

Original entry on oeis.org

1, 1, 3, 17, 137, 1457, 19355, 308961, 5766353, 123285153, 2972114803, 79782059249, 2360417058521, 76319622510289, 2677629295171979, 101318751122847297, 4113158120834726049, 178328823993199602241, 8223999403291995520995, 401989145900847087408849
Offset: 0

Views

Author

Paul D. Hanna, Nov 20 2012

Keywords

Comments

Compare to the LambertW identity: Sum_{n>=0} (n+1)^(n-1)*exp(-n*x)*x^n/n! = exp(x).

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 137*x^4/4! + 1457*x^5/5! +...
where
A(x) = 1 + sinh(x) + 3^1*sinh(x)^2/2! + 4^2*sinh(x)^3/3! + 5^3*sinh(x)^4/4! +...

Links

Crossrefs

Cf. A003724, A136630, A162650.
Cf. A238085.

Programs

  • Mathematica
    CoefficientList[Series[-LambertW[-Sinh[x]]/Sinh[x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 08 2013 *)
  • PARI
    {a(n)=n!*polcoeff(sum(k=0,n,(k+1)^(k-1)*sinh(x + x*O(x^n))^k/k!),n)}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f.: LambertW(-sinh(x)) / (-sinh(x)).
a(n) ~ (1+exp(2))^(1/4) * n^(n-1) / (exp(n-1) * log(exp(-1) +sqrt(1+exp(-2)))^(n-1/2)). - Vaclav Kotesovec, Jul 08 2013
a(n) = Sum_{k=0..n} (k+1)^(k-1) * A136630(n,k). - Seiichi Manyama, Feb 15 2025

A235134 Expansion of e.g.f. 1/(1 - sinh(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 19, 153, 1561, 19563, 289339, 4932273, 95258161, 2055639123, 49019157859, 1280056939593, 36329281202761, 1113449691889083, 36651273215389579, 1289577677407798113, 48299079453732363361, 1918528841276621473443, 80559757274836073592499
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 03 2014

Keywords

Comments

Generally, for e.g.f. 1/(1-sinh(p*x))^(1/p) we have a(n) ~ n! * p^n / (Gamma(1/p) * 2^(1/(2*p)) * n^(1-1/p) * (arcsinh(1))^(n+1/p)).

Links

Crossrefs

Cf. A006154, A235135.
Cf. A001147, A001586, A136630, A235131, A380155.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-Sinh[2*x])^(1/2), {x, 0, 20}], x] * Range[0, 20]!
  • PARI
    x='x+O('x^50); Vec(serlaplace(1/(sqrt(1-sinh(2*x))))) \\ G. C. Greubel, Apr 05 2017
  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*2^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

Formula

a(n) ~ n! * 2^(n-1/4) / (sqrt(Pi*n) * (log(1+sqrt(2)))^(n+1/2)).
a(n) = Sum_{k=0..n} A001147(k) * 2^(n-k) * A136630(n,k). - Seiichi Manyama, Jun 24 2025

A235128 Expansion of e.g.f. 1/(1 - sin(7*x))^(1/7).

Original entry on oeis.org

1, 1, 8, 71, 1072, 20161, 476288, 13315751, 432387712, 15959926081, 660372282368, 30265936565831, 1522069164439552, 83327826089289601, 4933286107483701248, 314052936209639958311, 21392225375507849838592, 1552501782546292090638721, 119588747474281844162428928
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 03 2014

Keywords

Comments

Generally, for e.g.f. 1/(1-sin(p*x))^(1/p) we have a(n) ~ n! * 2^(n+3/p) * p^n / (Gamma(2/p) * n^(1-2/p) * Pi^(n+2/p)).

Crossrefs

Cf. A001586 (p=2), A007788 (p=3), A144015 (p=4), A230134 (p=5), A227544 (p=6), A230114 (p=8).
Cf. A045754, A136630.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-Sin[7*x])^(1/7), {x, 0, 20}], x] * Range[0, 20]!
  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a045754(n) = prod(k=0, n-1, 7*k+1);
a(n) = sum(k=0, n, a045754(k)*(7*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

Formula

a(n) ~ n! * 2^(n+3/7) * 7^n / (Gamma(2/7) * n^(5/7) * Pi^(n+2/7)).
a(n) = Sum_{k=0..n} A045754(k) * (7*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A235135 Expansion of e.g.f. 1/(1 - sinh(3*x))^(1/3).

Original entry on oeis.org

1, 1, 4, 37, 424, 6241, 113824, 2460277, 61504384, 1746727201, 55545439744, 1955176596517, 75470959673344, 3169939381277761, 143927870364811264, 7024566555751464757, 366742587098140770304, 20394984041632355113921, 1203587891190987380752384, 75124090160952970927512997
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 03 2014

Keywords

Comments

Generally, for e.g.f. 1/(1-sinh(p*x))^(1/p) we have a(n) ~ n! * p^n / (Gamma(1/p) * 2^(1/(2*p)) * n^(1-1/p) * (arcsinh(1))^(n+1/p)).

Crossrefs

Cf. A006154, A235134.
Cf. A007559, A007788, A136630, A235132.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-Sinh[3*x])^(1/3), {x, 0, 20}], x] * Range[0, 20]!
  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*3^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

Formula

a(n) ~ n! * 3^n / (Gamma(1/3) * 2^(1/6) * n^(2/3) * (log(1+sqrt(2)))^(n+1/3)).
a(n) = Sum_{k=0..n} A007559(k) * 3^(n-k) * A136630(n,k). - Seiichi Manyama, Jun 24 2025

A352279 a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1).

Original entry on oeis.org

1, 2, 4, 10, 32, 114, 448, 1978, 9472, 48738, 270336, 1595114, 9965568, 65852882, 457326592, 3329243546, 25356271616, 201326396098, 1663597019136, 14279558011850, 127044810702848, 1170023757062450, 11136610150121472, 109395885009537402, 1107781178494025728, 11549900930966957346
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 10 2022

Keywords

Links

Crossrefs

Cf. A000557, A000807, A001861.
Cf. A003724, A136630, A352280.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, 2 k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 25}]
nmax = 25; CoefficientList[Series[Exp[2 Sinh[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k * Binomial[n, k] * BellB[k, -1] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 27 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*sinh(x)))) \\ Seiichi Manyama, Mar 26 2022

Formula

E.g.f.: exp( 2 * sinh(x) ).
a(n) = Sum_{k=0..n} 2^k * A136630(n,k). - Seiichi Manyama, Feb 18 2025

A381142 Expansion of e.g.f. exp( -LambertW(-sin(x)) ).

Original entry on oeis.org

1, 1, 3, 15, 113, 1137, 14355, 218239, 3883585, 79218721, 1822842243, 46717337007, 1319891043569, 40759239427857, 1365932381706963, 49373610759452575, 1914856819983977473, 79316216447375396161, 3494800326874932467331, 163218136611270923087439
Offset: 0

Views

Author

Seiichi Manyama, Feb 15 2025

Keywords

Links

Crossrefs

Cf. A002017, A381145, A381148.
Cf. A136630, A185690, A277498.

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+1)^(k-1)*I^(n-k)*a136630(n, k));

Formula

E.g.f. A(x) satisfies A(x) = exp( sin(x) * A(x) ).
a(n) = Sum_{k=0..n} (k+1)^(k-1) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

A381181 Expansion of e.g.f. (1/x) * Series_Reversion( x / (1 + sin(x)) ).

Original entry on oeis.org

1, 1, 2, 5, 8, -79, -1584, -20539, -223616, -1855295, -1736960, 435730789, 14511117312, 338965239601, 6202042886144, 71638247035109, -714560796196864, -84697775518956799, -3650903032332091392, -115829159202293866939, -2739961030150105333760, -29414406825401517785039
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2025

Keywords

Links

Crossrefs

Cf. A201627, A381180, A381182.
Cf. A162653, A381171, A381173.
Cf. A136630, A196776.

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(n+1, k)*I^(n-k)*a136630(n, k))/(n+1);

Formula

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

A381286 Expansion of e.g.f. 1/(1 - sin(3*x) / 3).

Original entry on oeis.org

1, 1, 2, -3, -48, -339, -1008, 10737, 237312, 2362041, 5432832, -318158523, -7615254528, -87216236379, 173049219072, 33959321252217, 851545449234432, 9561733579228401, -129701862228492288, -9445723672920941043, -239815723596207095808, -2109465061216228379619
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2025

Keywords

Crossrefs

Cf. A000111, A381285.
Cf. A136630, A352638, A381278.

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*I)^(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*i)^(n-k) * A136630(n,k), where i is the imaginary unit.

A381512 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = (2*n+k)!/k! * [x^(2*n+k)] sinh(x)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 10, 16, 1, 0, 1, 20, 91, 64, 1, 0, 1, 35, 336, 820, 256, 1, 0, 1, 56, 966, 5440, 7381, 1024, 1, 0, 1, 84, 2352, 24970, 87296, 66430, 4096, 1, 0, 1, 120, 5082, 90112, 631631, 1397760, 597871, 16384, 1, 0, 1, 165, 10032, 273988, 3331328, 15857205, 22368256, 5380840, 65536, 1, 0
Offset: 0

Views

Author

Seiichi Manyama, May 11 2025

Keywords

Examples

			Square array begins:
  1, 1,    1,     1,       1,        1, ...
  0, 1,    4,    10,      20,       35, ...
  0, 1,   16,    91,     336,      966, ...
  0, 1,   64,   820,    5440,    24970, ...
  0, 1,  256,  7381,   87296,   631631, ...
  0, 1, 1024, 66430, 1397760, 15857205, ...

Crossrefs

Columns k=0..7 give A000007, A000012, A000302, A002452(n+1), A166984, A002453, 4^n * A002451(n), A381513.
Main diagonal gives A383837.
Cf. A136630, A160562, A286899.

Programs

  • PARI
    a(n, k) = (2*n+k)!/k!*polcoef(sinh(x+x*O(x^(2*n+k)))^k, 2*n+k);

Formula

G.f. of column k: 1/Product_{j=0..floor(k/2)} (1 - (k-2*j)^2*x).
A(n,k) = k^2 * A(n-1,k) + A(n,k-2) for k > 1.
A(n,k) = (1/(2^k*k!)) * Sum_{j=0..k} (-1)^j * (k-2*j)^(2*n+k) * binomial(k,j).

A385304 Expansion of e.g.f. 1/(1 - 2 * sinh(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 16, 117, 1096, 12543, 169576, 2644617, 46735936, 922993083, 20145579136, 481555537917, 12511452674176, 351058439096823, 10579734482269696, 340820224678288017, 11687491783287586816, 425075150516293691763, 16343274366458168160256, 662325275389743380902917
Offset: 0

Views

Author

Seiichi Manyama, Jun 24 2025

Keywords

Crossrefs

Cf. A006154, A385305.
Cf. A380015, A385306, A385308, A385310.
Cf. A001147, A136630, A235134, A364822.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} A001147(k) * A136630(n,k).
a(n) ~ sqrt(2) * n^n / (5^(1/4) * exp(n) * log((1 + sqrt(5))/2)^(n + 1/2)). - Vaclav Kotesovec, Jun 28 2025
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