A006531 -id:A006531 - cp's OEIS frontend

A087138 Expansion of (1-sqrt(1-4*log(1+x)))/2.

1, 1, 8, 64, 824, 12968, 252720, 5789712, 153169440, 4589004192, 153643615872, 5684390364288, 230307823878144, 10141452865049088, 482259966649655808, 24630247225278881280, 1344614199041549399040, 78137673004382654223360

Offset: 1

Vladeta Jovovic, Oct 18 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 1..372
Index entries for reversions of series

Cf. A000108, A052851, A052803, A052886, A052895, A006531, A048287, A293604.

Cf. A370462, A370463.

Rest[CoefficientList[Series[(1-Sqrt[1-4*Log[1+x]])/2, {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, May 03 2015 *)

x='x+O('x^50); Vec(serlaplace((1-sqrt(1-4*log(1+x)))/2)) \\ G. C. Greubel, May 24 2017

a(n) = Sum_{k=1..n} Stirling1(n, k)*k!*Catalan(k-1).
a(n) ~ n! / (2*exp(1/8)*sqrt(Pi) * (exp(1/4)-1)^(n-1/2) * n^(3/2)). - Vaclav Kotesovec, May 03 2015
From Seiichi Manyama, Sep 09 2024: (Start)
E.g.f. satisfies A(x) = (log(1 + x)) / (1 - A(x)).
E.g.f.: Series_Reversion( exp(x * (1 - x)) - 1 ). (End)

A087152 Expansion of (1-sqrt(1-4*log(1+x)))/log(1+x)/2.

Original entry on oeis.org

1, 3, 20, 194, 2554, 42226, 843744, 19769256, 531768120, 16152296424, 546895099200, 20425461026736, 834215500905552, 36988602430554576, 1769524998544143360, 90851799797294235264, 4982968503277896871296

Offset: 1

Vladeta Jovovic, Oct 18 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 1..370

Cf. A000108, A052851, A052803, A052886, A052895, A006531, A048287.

Rest[CoefficientList[Series[(1-Sqrt[1-4*Log[1+x]])/Log[1+x]/2, {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, May 03 2015 *)

x='x+O('x^50); Vec(serlaplace((1-sqrt(1-4*log(1+x)))/log(1+x)/2 - 1)) \\ G. C. Greubel, May 24 2017

a(n) = Sum_{k=0..n} Stirling1(n, k)*k!*Catalan(k).

a(n) ~ 2*n! / (exp(1/8)*sqrt(Pi) * (exp(1/4)-1)^(n-1/2) * n^(3/2)). - Vaclav Kotesovec, May 03 2015

A295239 Expansion of e.g.f. 2/(1 + sqrt(1 + 4xexp(x))).

Original entry on oeis.org

1, -1, 2, -9, 68, -705, 9234, -146209, 2717000, -57986433, 1397949830, -37576332321, 1114326129564, -36141571087297, 1272713716466906, -48360394499269665, 1972269941821097744, -85929979225787811585, 3983422470176606823054, -195765982110500512129057

Offset: 0

Ilya Gutkovskiy, Nov 18 2017

sign

Cf. A000108, A006531, A052895, A295238.

a:=series(2/(1+sqrt(1+4*x*exp(x))),x=0,20): seq(n!*coeff(a,x,n),n=0..19); # Paolo P. Lava, Mar 27 2019

nmax = 19; CoefficientList[Series[2/(1 + Sqrt[1 + 4 x Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[x Exp[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n, k] k! Sum[(-1)^m (m + 1)^(k - m - 1) Binomial[2 m, m]/(k - m)!, {m, 0, k}], {k, 0, n}], {n, 0, 19}]

a(n) = n!*sum(k=0, n, (-1)^k*k^(n-k)*binomial(2*k, k)/((k+1)*(n-k)!)); \\ Seiichi Manyama, Oct 30 2024

E.g.f.: 1/(1 + x*exp(x)/(1 + x*exp(x)/(1 + x*exp(x)/(1 + x*exp(x)/(1 + ...))))), a continued fraction.
a(n) ~ sqrt(2*(1+LambertW(-1/4))) * n^(n-1) / (exp(n) * (LambertW(-1/4))^n). - Vaclav Kotesovec, Nov 18 2017
a(n) = n! * Sum_{k=0..n} (-1)^k * k^(n-k) * A000108(k)/(n-k)!. - Seiichi Manyama, Oct 30 2024

A367164 E.g.f. satisfies A(x) = 1 + A(x)^3 * (1 - exp(-x)).

Original entry on oeis.org

1, 1, 5, 55, 929, 21271, 616265, 21624415, 891671009, 42263854471, 2264336600825, 135325966276975, 8926057815521489, 644116254555006871, 50477965058305364585, 4269330999037434100735, 387619447676360230226369, 37602089272441407334114471

Offset: 0

Seiichi Manyama, Nov 07 2023

nonn

Cf. A367155, A367158, A367161.

Cf. A006531.

Table[Sum[(-1)^(n-k) * (3*k)!/(2*k+1)! * StirlingS2[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 10 2023 *)

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

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

a(n) ~ sqrt(69) * n^(n-1) / (2^(5/2) * log(27/23)^(n - 1/2) * exp(n)). - Vaclav Kotesovec, Nov 10 2023

A355290 a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling2(n,k) * Catalan(k).

Original entry on oeis.org

1, 1, 1, 0, -3, -2, 23, 17, -333, 86, 6941, -17025, -160267, 1082864, 2273807, -56742606, 152154285, 2293098332, -22007462809, -15179437171, 1671107690083, -10716783889040, -58404948615167, 1439391012463810, -6701658223127029, -88340107011433060

Offset: 0

Seiichi Manyama, Jun 27 2022

sign

Cf. A000108, A006531, A064856, A086662, A086672.

A355290 := proc(n)
    add((-1)^(n-k)*stirling2(n,k)*A000108(k),k=0..n) ;
end proc:
seq(A355290(n),n=0..70) ; # R. J. Mathar, Mar 13 2023

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

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, binomial(2*k, k)/(k+1)*x^k/prod(j=1, k, 1+j*x)))

G.f.: Sum_{k>=0} Catalan(k) * x^k / Product_{j=1..k} (1 + j*x).

A069657 Sum( S(n,k) * M(k-1), k=1..n), where S(n,k) = Stirling numbers of the second kind, M(n) = Motzkin numbers, A001006.

Original entry on oeis.org

0, 1, 2, 6, 24, 115, 628, 3818, 25455, 183968, 1428184, 11824098, 103794727, 961461179, 9360372700, 95448502365, 1016413911387, 11273822075837, 129950445723958, 1553488011957986, 19225242250821071, 245899882175001580, 3245812116097119188, 44155099624566615247

Offset: 0

N. J. A. Sloane, Jun 06 2002

nonn

Cf. A006531, A001006.

A335748 T(n,k) = (-1)^n(binomial(2k,k)/(k+1))Sum_{j=0..n} (-1)^jbinomial(k,j)*j^n. Triangle read by rows, T(n, k) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, -1, 4, 0, 1, -12, 30, 0, -1, 28, -180, 336, 0, 1, -60, 750, -3360, 5040, 0, -1, 124, -2700, 21840, -75600, 95040, 0, 1, -252, 9030, -117600, 705600, -1995840, 2162160, 0, -1, 508, -28980, 571536, -5292000, 25280640, -60540480, 57657600

Offset: 0

Peter Luschny, Jul 09 2020

			                             [0] 1
                           [1] 0, 1
                         [2] 0, -1, 4
                       [3] 0, 1, -12, 30
                   [4] 0, -1, 28, -180, 336
                [5] 0, 1, -60, 750, -3360, 5040
          [6] 0, -1, 124, -2700, 21840, -75600, 95040
   [7] 0, 1, -252, 9030, -117600, 705600, -1995840, 2162160
[8] 0, -1, 508, -28980, 571536, -5292000, 25280640, -60540480, 57657600

Cf. A006531 (row sums), A052895 (absolute row sums), T(n,n) = A001761(n) (signed A292220(n)).

T(n, k) = (-1)^n*CatalanNumber(k)*Sum_{j=0..n}(-1)^j*binomial(k, j)*j^n.

cp's OEIS Frontend

A087138 Expansion of (1-sqrt(1-4*log(1+x)))/2.

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A087152 Expansion of (1-sqrt(1-4*log(1+x)))/log(1+x)/2.

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A295239 Expansion of e.g.f. 2/(1 + sqrt(1 + 4*x*exp(x))).

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A367164 E.g.f. satisfies A(x) = 1 + A(x)^3 * (1 - exp(-x)).

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A355290 a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling2(n,k) * Catalan(k).

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A069657 Sum( S(n,k) * M(k-1), k=1..n), where S(n,k) = Stirling numbers of the second kind, M(n) = Motzkin numbers, A001006.

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A335748 T(n,k) = (-1)^n*(binomial(2*k,k)/(k+1))*Sum_{j=0..n} (-1)^j*binomial(k,j)*j^n. Triangle read by rows, T(n, k) for n >= 0 and 0 <= k <= n.

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A295239 Expansion of e.g.f. 2/(1 + sqrt(1 + 4xexp(x))).

A335748 T(n,k) = (-1)^n(binomial(2k,k)/(k+1))Sum_{j=0..n} (-1)^jbinomial(k,j)*j^n. Triangle read by rows, T(n, k) for n >= 0 and 0 <= k <= n.