A052848 -id:A052848 - cp's OEIS frontend

A375588 Expansion of e.g.f. 1 / (1 + x - x * exp(x^2)).

1, 0, 0, 6, 0, 60, 720, 840, 40320, 378000, 2116800, 60207840, 598752000, 7792424640, 181863601920, 2288689603200, 45855781171200, 1016682053587200, 17113328962329600, 422970486434496000, 9765438564930048000, 213305542403822668800, 5916931500898517299200

Offset: 0

Seiichi Manyama, Aug 19 2024

nonn

Cf. A052848, A375589.

Cf. A357966, A375561.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x-x*exp(x^2))))

a(n) = n!*sum(k=0, n\2, (n-2*k)!*stirling(k, n-2*k, 2)/k!);

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)! * Stirling2(k,n-2*k)/k!.

A375660 Expansion of e.g.f. 1 / (1 - x * (exp(x) - 1))^2.

Original entry on oeis.org

1, 0, 4, 6, 80, 370, 4152, 34034, 413632, 4744674, 66354680, 954512482, 15454225536, 263909265074, 4898255210968, 96284064551250, 2022022344889472, 44858682139345090, 1052826609589372152, 25994393541984673154, 674563101823606851520, 18337775305498096349202

Offset: 0

Seiichi Manyama, Aug 23 2024

nonn

Cf. A052848, A375661.

Cf. A005649.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x*(exp(x)-1))^2))

a(n) = n!*sum(k=0, n\2, (k+1)!*stirling(n-k, k, 2)/(n-k)!);

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A052848.

a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)! * Stirling2(n-k,k)/(n-k)!.

A375661 Expansion of e.g.f. 1 / (1 - x * (exp(x) - 1))^3.

Original entry on oeis.org

1, 0, 6, 9, 156, 735, 9738, 83181, 1129656, 13662459, 207281190, 3151269033, 54457383060, 980680471095, 19240001086530, 397345461622245, 8763618490102128, 203472380293912563, 4991552271140255838, 128517790560854181537, 3472936316648987980620

Offset: 0

Seiichi Manyama, Aug 23 2024

nonn

Cf. A052848, A375660.

Cf. A226515.

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

a(n) = n!*sum(k=0, n\2, (k+2)!*stirling(n-k, k, 2)/(n-k)!)/2;

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A052848.

a(n) = (n!/2) * Sum_{k=0..floor(n/2)} (k+2)! * Stirling2(n-k,k)/(n-k)!.

A354313 Expansion of e.g.f. 1/(1 - x/2 * (exp(2 * x) - 1)).

Original entry on oeis.org

1, 0, 2, 6, 40, 280, 2496, 25424, 297984, 3920256, 57349120, 922611712, 16193375232, 307896882176, 6304666798080, 138318662000640, 3236895083167744, 80483201605795840, 2118875812456366080, 58882581280649117696, 1722441885524719042560

Offset: 0

Seiichi Manyama, May 23 2022

nonn

Cf. A052848, A354314.

Cf. A216794, A353998, A354311.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x/2*(exp(2*x)-1))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*2^(j-2)*binomial(i, j)*v[i-j+1])); v;

a(n) = n!*sum(k=0, n\2, 2^(n-2*k)*k!*stirling(n-k, k, 2)/(n-k)!);

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

a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-2*k) * k! * Stirling2(n-k,k)/(n-k)!.

A367880 Expansion of e.g.f. 1/(1 - 2 * x * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 4, 6, 104, 490, 7452, 65534, 1062224, 13825746, 252414020, 4303920742, 89701635960, 1870259792570, 44391086228972, 1085906907998670, 29112549152845472, 813723252665063842, 24402507959486170260, 765358519469125339190

Offset: 0

Seiichi Manyama, Dec 03 2023

nonn

Cf. A052848, A367881.

a(n) = n!*sum(k=0, n\2, 2^k*k!*stirling(n-k, k, 2)/(n-k)!);

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

a(n) = n! * Sum_{k=0..floor(n/2)} 2^k * k! * Stirling2(n-k,k)/(n-k)!.

A353994 Expansion of e.g.f. 1/(1 + log(1 - x) * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 2, 6, 42, 305, 2815, 29792, 362432, 4952481, 75239143, 1257202584, 22918653428, 452620972245, 9626556838015, 219367419292972, 5332164894151648, 137709755844024929, 3765736630207259055, 108696751776637007080, 3302628833563666988740

Offset: 0

Seiichi Manyama, May 13 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..421

Cf. A052830, A052848, A052863, A305407, A353995.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)*(exp(x)-1))))

a(0) = 1; a(n) = Sum_{k=1..n} A052863(k) * binomial(n,k) * a(n-k).

A354314 Expansion of e.g.f. 1/(1 - x/3 * (exp(3 * x) - 1)).

Original entry on oeis.org

1, 0, 2, 9, 60, 495, 4986, 58401, 780984, 11749779, 196446870, 3612882933, 72484364052, 1575418827879, 36875093680530, 924769734574185, 24737895033896304, 703105981990977915, 21159355356941587470, 672148402091190649629, 22475238194908656800460

Offset: 0

Seiichi Manyama, May 23 2022

nonn

Cf. A052848, A354313.

Cf. A288834, A328182, A353999, A354312.

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

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*3^(j-2)*binomial(i, j)*v[i-j+1])); v;

a(n) = n!*sum(k=0, n\2, 3^(n-2*k)*k!*stirling(n-k, k, 2)/(n-k)!);

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

a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-2*k) * k! * Stirling2(n-k,k)/(n-k)!.

A371142 E.g.f. satisfies A(x) = 1 + xA(x)^2 (exp(x) - 1).

Original entry on oeis.org

1, 0, 2, 3, 52, 245, 4446, 40327, 823432, 11539593, 269030530, 5151458411, 137198342700, 3328477618765, 100753779655270, 2948537330575215, 100684293424304656, 3437897124964763921, 131334298073705194506, 5110990692709364143219, 216634434142329948020020

Offset: 0

Seiichi Manyama, Mar 12 2024

nonn

Cf. A052848, A371143.

Cf. A371140.

nmax = 20; CoefficientList[Series[(1 - Sqrt[1 + 4*x - 4*E^x*x]) / (2*(-x + E^x*x)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 12 2024 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(2/(1+sqrt(1-4*x*(exp(x)-1)))))

a(n) = n!*sum(k=0, n\2, (2*k)!/(k+1)!*stirling(n-k, k, 2)/(n-k)!);

E.g.f.: 2/(1 + sqrt(1-4*x*(exp(x)-1))).
a(n) = n! * Sum_{k=0..floor(n/2)} (2*k)!/(k+1)! * Stirling2(n-k,k)/(n-k)!.
a(n) ~ sqrt(2 + 2*r + 8*r^2) * n^(n-1) / (exp(n) * r^n), where r = 0.4454633431717774246390371283449007888959504623839... is the positive real root of the equation 4*r*(exp(r) - 1) = 1. - Vaclav Kotesovec, Mar 12 2024

A371143 E.g.f. satisfies A(x) = 1 + xA(x)^3 (exp(x) - 1).

Original entry on oeis.org

1, 0, 2, 3, 76, 365, 9906, 94507, 2832824, 43209945, 1438766830, 30971280791, 1146868043124, 32166137748901, 1322928667341386, 45791799761422275, 2085517396191903856, 85748423669245738673, 4306944218393176448742, 204597526239295278145327

Offset: 0

Seiichi Manyama, Mar 12 2024

nonn

Cf. A052848, A371142.

Cf. A371141.

a(n) = n!*sum(k=0, n\2, (3*k)!/(2*k+1)!*stirling(n-k, k, 2)/(n-k)!);

a(n) = n! * Sum_{k=0..floor(n/2)} (3*k)!/(2*k+1)! * Stirling2(n-k,k)/(n-k)!.

A375586 Expansion of e.g.f. 1 / (1 + x - x * exp(x^2/2)).

Original entry on oeis.org

1, 0, 0, 3, 0, 15, 180, 105, 5040, 46305, 132300, 3752595, 33679800, 243378135, 5565940380, 56191160025, 712410098400, 14889814164225, 183558878603100, 3236148386145675, 66650136566013000, 1027807726886515575, 21983938825036488300, 469896981350215644225

Offset: 0

Seiichi Manyama, Aug 19 2024

nonn

Cf. A052848, A375587.

Cf. A375167.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x-x*exp(x^2/2))))

a(n) = n!*sum(k=0, n\2, (n-2*k)!*stirling(k, n-2*k, 2)/(2^k*k!));

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)! * Stirling2(k,n-2*k)/(2^k*k!).

cp's OEIS Frontend

A375588 Expansion of e.g.f. 1 / (1 + x - x * exp(x^2)).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A375660 Expansion of e.g.f. 1 / (1 - x * (exp(x) - 1))^2.

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A375661 Expansion of e.g.f. 1 / (1 - x * (exp(x) - 1))^3.

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A354313 Expansion of e.g.f. 1/(1 - x/2 * (exp(2 * x) - 1)).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

PARI

Formula

A367880 Expansion of e.g.f. 1/(1 - 2 * x * (exp(x) - 1)).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A353994 Expansion of e.g.f. 1/(1 + log(1 - x) * (exp(x) - 1)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A354314 Expansion of e.g.f. 1/(1 - x/3 * (exp(3 * x) - 1)).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

PARI

Formula

A371142 E.g.f. satisfies A(x) = 1 + x*A(x)^2 * (exp(x) - 1).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A371143 E.g.f. satisfies A(x) = 1 + x*A(x)^3 * (exp(x) - 1).

Views

Author

Keywords

Crossrefs

A371142 E.g.f. satisfies A(x) = 1 + xA(x)^2 (exp(x) - 1).

A371143 E.g.f. satisfies A(x) = 1 + xA(x)^3 (exp(x) - 1).