A006154 -id:A006154 - cp's OEIS frontend

A322623 E.g.f.: (1 + sinh(x)) / (1 - sinh(x)).

1, 2, 4, 14, 64, 362, 2464, 19574, 177664, 1814162, 20583424, 256891934, 3497611264, 51588733562, 819450793984, 13946142745094, 253171058212864, 4883182404118562, 99727612182790144, 2149854113300939054, 48784173816258494464, 1162353473295706049162, 29013549746780744187904, 757126891483681641073814, 20616734677807356197208064, 584789894473832421848925362

Offset: 0

Paul D. Hanna, Dec 29 2018

nonn

Equals the antidiagonal sums of square table A322620.

a(n) = 2*A006154(n) for n >= 1.

			E.g.f.: A(x) = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 64*x^4/4! + 362*x^5/5! + 2464*x^6/6! + 19574*x^7/7! + 177664*x^8/8! + 1814162*x^9/9! + ...
where
A(x) = 1 + 2*sinh(x) + 2*sinh(x)^2 + 2*sinh(x)^3 + 2*sinh(x)^4 + ...

Robert Israel, Table of n, a(n) for n = 0..440

Cf. A322620, A012261, A006154.

S:= series((1+sinh(x))/(1-sinh(x)),x,51):
seq(coeff(S,x,j)*j!,j=0..50);  #  Robert Israel, Dec 31 2018

{a(n) = my(X = x +x*O(x^n)); n! * polcoeff( (1 + sinh(X)) / (1 - sinh(X)),n)}
for(n=0,30, print1(a(n),", "))

a(n) = Sum_{k=0..n} A322620(n-k,k), for n >= 0.

a(n) ~ sqrt(2)*n!/log(1+sqrt(2))^(n+1). - Robert Israel, Dec 31 2018

A330046 Expansion of e.g.f. exp(x) / (1 - sinh(x)).

Original entry on oeis.org

1, 2, 5, 17, 77, 437, 2975, 23627, 214457, 2189897, 24846395, 310095887, 4221990437, 62273111357, 989164604615, 16834483468547, 305604501324017, 5894522593612817, 120381876933435635, 2595103478745235607, 58887707028270711197, 1403084759749993342277

Offset: 0

Ilya Gutkovskiy, Nov 28 2019

nonn

Binomial transform of A006154.

Cf. A000629, A006154, A009283, A327034, A330047.

nmax = 21; CoefficientList[Series[Exp[x]/(1 - Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = Sum_{k=0..n} binomial(n,k) * A006154(k).

a(n) ~ n! * (1 + 1/sqrt(2)) / (log(1 + sqrt(2)))^(n+1). - Vaclav Kotesovec, Dec 03 2019

A332258 E.g.f.: 1 / (1 + x - sinh(x)).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 20, 1, 112, 1681, 492, 27721, 371624, 319177, 13461604, 171387217, 319071456, 11466038689, 143550642140, 484491620089, 15758152572952, 199089883272217, 1077471975974484, 32827750137627457, 427744154995090256, 3385134777669637681

Offset: 0

Ilya Gutkovskiy, Feb 08 2020

nonn

Number of labeled ordered partitions of an n-set into odd parts > 1.

Cf. A006154, A009227, A032032.

nmax = 25; CoefficientList[Series[1/(1 + x - Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 2 k - 1] a[n - 2 k + 1], {k, 2, Ceiling[n/2]}]; Table[a[n], {n, 0, 25}]

seq(n)={Vec(serlaplace(1 / (1 + x - sinh(x + O(x*x^n)))))} \\ Andrew Howroyd, Feb 08 2020

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

a(n) ~ n! / ((cosh(r) - 1) * r^(n+1)), where r = 1.72911689821437486498840709347... is the root of the equation 1 + r - sinh(r) = 0. - Vaclav Kotesovec, Feb 08 2020

A349105 Expansion of e.g.f. 1/(1 - (sinh(x) + x*cosh(x))/2 ).

Original entry on oeis.org

1, 1, 2, 8, 40, 243, 1796, 15502, 152608, 1690613, 20814208, 281859540, 4163795648, 66636761575, 1148477490304, 21207704998010, 417728195909632, 8742243282090153, 193720478508563456, 4531158728871170080, 111562803180301643776, 2884156736234559267611

Offset: 0

Seiichi Manyama, Mar 26 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..435

Cf. A006154, A205571, A349104, A351937.

With[{m = 21}, Range[0, m]! * CoefficientList[Series[1/(1 - (Sinh[x] + x*Cosh[x])/2), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-(sinh(x)+x*cosh(x))/2)))

a(n) = if(n==0, 1, sum(k=0, (n-1)\2, (k+1)*binomial(n, 2*k+1)*a(n-2*k-1)));

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

A352251 Expansion of e.g.f. 1 / (1 - x * sinh(x)) (even powers only).

Original entry on oeis.org

1, 2, 28, 966, 62280, 6452650, 980531916, 205438870014, 56760128400016, 19994672935658322, 8746764024725937300, 4651991306703670964518, 2956156902003429777549144, 2212026607642404922284728826, 1925137044528752884360406444380, 1928103808741894922401976601295950

Offset: 0

Ilya Gutkovskiy, Mar 09 2022

nonn

Cf. A006153, A006154, A009233, A094088, A205571, A210672, A351762, A352250, A352252, A352253.

nmax = 30; Take[CoefficientList[Series[1/(1 - x Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[2 n, 2 k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]

my(x='x+O('x^40), v=Vec(serlaplace(1 /(1-x*sinh(x))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

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

A352326 Expansion of e.g.f.: 1/(2 - exp(x) - sinh(x)).

Original entry on oeis.org

1, 2, 9, 62, 567, 6482, 88929, 1423382, 26037027, 535813802, 12251630349, 308153112302, 8455276083087, 251333936555522, 8045613346221369, 275950004166050822, 10095559110771678747, 392427366313299119642, 16151459739717643489989

Offset: 0

Seiichi Manyama, Mar 12 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..395

Cf. A000670, A006154, A028296, A094088, A352327, A352617.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-k)*binomial(n, k)*(1+(k mod 2)), k=1..n))
    end:
seq(a(n), n=0..18); # Alois P. Heinz, Mar 25 2022

m = 18; Range[0, m]! * CoefficientList[Series[1/(2 - Exp[x] - Sinh[x]), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(x)-sinh(x))))

a(n) = if(n==0, 1, sum(k=1, n, (3-(-1)^k)/2*binomial(n, k)*a(n-k)));

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

a(n) ~ n! / (sqrt(7) * log((2 + sqrt(7))/3)^(n+1)). - Vaclav Kotesovec, Mar 12 2022

A352327 Expansion of e.g.f.: 1/(3 - exp(x) - cosh(x)).

Original entry on oeis.org

1, 1, 4, 19, 130, 1081, 10894, 127639, 1711210, 25798141, 432212134, 7964801659, 160121522290, 3487254825601, 81790592435374, 2055350489070079, 55093108433421370, 1569052795651631461, 47315282424232826614, 1506074331671551028899

Offset: 0

Seiichi Manyama, Mar 12 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..414

Cf. A000670, A006154, A094088, A352326, A352624.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-k)*binomial(n, k)*(2-(k mod 2)), k=1..n))
    end:
seq(a(n), n=0..19); # Alois P. Heinz, Mar 25 2022

m = 19; Range[0, m]! * CoefficientList[Series[1/(3 - Exp[x] - Cosh[x]), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(3-exp(x)-cosh(x))))

a(n) = if(n==0, 1, sum(k=1, n, (3+(-1)^k)/2*binomial(n, k)*a(n-k)));

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

a(n) ~ n! / (sqrt(6) * log(1 + sqrt(2/3))^(n+1)). - Vaclav Kotesovec, Mar 12 2022

A352428 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n,3k+1) a(n-3*k-1).

Original entry on oeis.org

1, 1, 2, 6, 25, 130, 810, 5881, 48806, 455706, 4727881, 53955682, 671730246, 9059714665, 131588822822, 2047796305470, 33992509701721, 599526848094850, 11195864285933682, 220692569175568729, 4579248276057441926, 99767702172338210898, 2277136869014579978473, 54336724559407913237122

Offset: 0

Ilya Gutkovskiy, Mar 16 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..447

Cf. A000670, A006154, A243664, A291973, A333881, A352429, A352430.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 3 k + 1] a[n - 3 k - 1], {k, 0, Floor[(n - 1)/3]}]; Table[a[n], {n, 0, 23}]
nmax = 23; CoefficientList[Series[1/(1 - Sum[x^(3 k + 1)/(3 k + 1)!, {k, 0, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\3, x^(3*k+1)/(3*k+1)!)))) \\ Seiichi Manyama, Mar 23 2022

E.g.f.: 1 / (1 - Sum_{k>=0} x^(3*k+1) / (3*k+1)!).

A352430 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/5)} binomial(n,5k+1) a(n-5*k-1).

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 721, 5054, 40488, 364896, 3654000, 40249441, 483659508, 6296246424, 88269037584, 1325861901000, 21243052172161, 361630022931666, 6518319228715302, 124018898163736536, 2483799332459535000, 52231733840672804881, 1150683180739820615582, 26502219276887376327696

Offset: 0

Ilya Gutkovskiy, Mar 16 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A000670, A006154, A243666, A333882, A352428, A352429.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 5 k + 1] a[n - 5 k - 1], {k, 0, Floor[(n - 1)/5]}]; Table[a[n], {n, 0, 23}]
nmax = 23; CoefficientList[Series[1/(1 - Sum[x^(5 k + 1)/(5 k + 1)!, {k, 0, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\5, x^(5*k+1)/(5*k+1)!)))) \\ Seiichi Manyama, Mar 23 2022

E.g.f.: 1 / (1 - Sum_{k>=0} x^(5*k+1) / (5*k+1)!).

A365915 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(2k+5) / (2k+5)! ).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 252, 1, 1584, 1, 7436, 756757, 31616, 14702689, 129404, 189559657, 11733266992, 2062481617, 516242875084, 20611819933, 14135172627712, 623557476714481, 312148517693820, 52096977907924561, 6121122865591920

Offset: 0

Seiichi Manyama, Sep 23 2023

Seiichi Manyama, Table of n, a(n) for n = 0..529

Cf. A245790, A365916, A365917.
Cf. A006154, A332258.
Cf. A365896.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x+x^3/6-sinh(x))))

a(0) = 1; a(n) = Sum_{k=0..floor((n-5)/2)} binomial(n,2*k+5) * a(n-2*k-5).

E.g.f.: 1 / ( 1 + x + x^3/6 - sinh(x) ).

cp's OEIS Frontend

A322623 E.g.f.: (1 + sinh(x)) / (1 - sinh(x)).

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A330046 Expansion of e.g.f. exp(x) / (1 - sinh(x)).

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A332258 E.g.f.: 1 / (1 + x - sinh(x)).

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A349105 Expansion of e.g.f. 1/(1 - (sinh(x) + x*cosh(x))/2 ).

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A352251 Expansion of e.g.f. 1 / (1 - x * sinh(x)) (even powers only).

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A352326 Expansion of e.g.f.: 1/(2 - exp(x) - sinh(x)).

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A352327 Expansion of e.g.f.: 1/(3 - exp(x) - cosh(x)).

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A352428 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n,3*k+1) * a(n-3*k-1).

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A352428 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n,3k+1) a(n-3*k-1).

A352430 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/5)} binomial(n,5k+1) a(n-5*k-1).

A365915 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(2k+5) / (2k+5)! ).