A048287 -id:A048287 - cp's OEIS frontend

A052851 Expansion of e.g.f. 1/2 - (1/2)(1+4log(1-x))^(1/2).

0, 1, 3, 20, 220, 3424, 69008, 1706256, 49956240, 1689497376, 64799254752, 2778906776832, 131756614920192, 6843405231815424, 386414606189283072, 23567401521343170048, 1543994621969805135360, 108137637714495023354880, 8062825821198926369725440

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

Seiichi Manyama, Table of n, a(n) for n = 0..360
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 819
Index entries for reversions of series

Cf. A048287, A052886, A087138.
Cf. A371314, A371315.
Cf. A371327, A376041, A376042.

spec := [S,{B=Cycle(Z),S=Prod(B,C),C=Sequence(S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[1/2-1/2*(1+4*Log[1-x])^(1/2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

a(n):=sum(stirling1(n,k)*k!*binomial(2*k-2,k-1)/k*(-1)^(n+k), k,1,n); /* Vladimir Kruchinin, May 12 2012 */

E.g.f.: 1/2 - (1/2)*(1-4*log(-1/(-1+x)))^(1/2).
a(n) = Sum_{k=1..n} Stirling1(n,k)*k!*C(2*k-2,k-1)/k*(-1)^(n+k). - Vladimir Kruchinin, May 12 2012
a(n) ~ n^(n-1)/(sqrt(2)*exp(3*n/4)*(exp(1/4)-1)^(n-1/2)). - Vaclav Kotesovec, Sep 30 2013
From Seiichi Manyama, Sep 09 2024: (Start)
E.g.f. satisfies A(x) = (-log(1 - x)) / (1 - A(x)).
E.g.f.: Series_Reversion( 1 - exp(-x * (1 - x)) ). (End)

New name using e.g.f., Vaclav Kotesovec, Sep 30 2013

A136595 Matrix inverse of triangle A136590.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 7, -3, 1, 0, -61, 31, -6, 1, 0, 751, -375, 85, -10, 1, 0, -11821, 5911, -1350, 185, -15, 1, 0, 226927, -113463, 26341, -3710, 350, -21, 1, 0, -5142061, 2571031, -603246, 87381, -8610, 602, -28, 1, 0, 134341711, -67170855, 15887845, -2346330, 240051, -17766

Offset: 0

Paul D. Hanna, Jan 10 2008

A136590 is the triangle of trinomial logarithmic coefficients.
Column 1 is signed A048287, which is the number of semiorders on n labeled nodes whose incomparability graph is connected.
The Bell transform of (-1)^n*A048287(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			Triangle begins:
1;
0, 1;
0, -1, 1;
0, 7, -3, 1;
0, -61, 31, -6, 1;
0, 751, -375, 85, -10, 1;
0, -11821, 5911, -1350, 185, -15, 1;
0, 226927, -113463, 26341, -3710, 350, -21, 1;
0, -5142061, 2571031, -603246, 87381, -8610, 602, -28, 1;
0, 134341711, -67170855, 15887845, -2346330, 240051, -17766, 966, -36, 1; ...

Cf. columns: A048287, A136596, A136597; A136590 (matrix inverse); A136588, A136589.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> (-1)^n*A048287(n+1), 9); # Peter Luschny, Jan 27 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 11;
M = BellMatrix[Sum[(-1)^(k+1) k! StirlingS2[#+1, k] CatalanNumber[k-1], {k, 1, #+1}]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

PARI
```
{T(n,k) = if(n
				
```

/* Define Stirling2: */
{Stirling2(n,k) = n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!,n)}
/* Define Catalan(m,n) = [x^n] C(x)^m: */
{CATALAN(m,n) = binomial(2*n+m,n) * m/(2*n+m)}
/* Define this triangle: */
{T(n,k) = if(n

# uses[bell_matrix from A264428]
bell_matrix(lambda n: (-1)^n*A048287(n+1), 10) # Peter Luschny, Jan 18 2016

T(n,k) = Sum_{i=0..n-1} (-1)^i * (k+i)! * Stirling2(n,k+i) * Catalan(k,i)/k!, where Stirling2(n,k) = A008277(n,k); Catalan(k,i) = C(2i+k,i)*k/(2i+k) = coefficient of x^i in C(x)^k with C(x) = (1-sqrt(1-4x))/(2x).

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

Original entry on oeis.org

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)

A136596 Column 2 of triangle A136595.

Original entry on oeis.org

1, -3, 31, -375, 5911, -113463, 2571031, -67170855, 1987919671, -65731585623, 2401646633431, -96089053104135, 4178215255335031, -196193483904124983, 9894077286353278231, -533334378459657706215, 30602112192036616407991

Offset: 2

Paul D. Hanna, Jan 10 2008

sign

Cf. A136595; A048287, A136597.

{a(n)=n!* sum(i=0,n-1,(-1)^i*polcoeff(((exp(x+x*O(x^n))-1)^(2+i)),n)*binomial(2*i+2,i)/(2*i+2))}
for(n=2,20,print1(a(n),", "))

/* Define Stirling2: */
{Stirling2(n,k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!,n)}
/* Define Catalan(m,n) = [x^n] C(x)^m: */
{Catalan(m,n)=binomial(2*n+m,n)*m/(2*n+m)}
/* Define this sequence: */
{a(n)=sum(i=0,n-1,(-1)^i*(2+i)!*Stirling2(n,2+i)*Catalan(2,i)/2!)}
for(n=2,20,print1(a(n),", "))

a(n) = Sum_{i=0..n-1} (-1)^i*(2+i)!*Stirling2(n,2+i)*Catalan(2,i)/2!, where Stirling2(n,k) = A008277(n,k); Catalan(k,i) = binomial(2*i+k,i)*k/(2*i+k) = coefficient of x^i in C(x)^k with C(x) = (1-sqrt(1-4x))/(2x).

a(n) = (1+(-1)^n*A048287(n))/2. - Vladeta Jovovic, Jan 27 2008

A136597 Column 3 of triangle A136595.

Original entry on oeis.org

1, -6, 85, -1350, 26341, -603246, 15887845, -473148150, 15723174181, -576826897086, 23157022930405, -1009818279438150, 47533643556874021, -2402218856253008526, 129730266330534913765, -7455932648513351731350, 454377365410347843373861

Offset: 3

Paul D. Hanna, Jan 10 2008

sign

Cf. A136595, A048287, A136596.

a(n)=n!/2!* sum(i=0,n-1,(-1)^i*polcoeff(((exp(x+x*O(x^n))-1)^(3+i)),n)*binomial(2*i+3,i)/(2*i+3))

/* Define Stirling2: */ {Stirling2(n,k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!,n)} /* Define Catalan(m,n) = [x^n] C(x)^m: */ {Catalan(m,n)=binomial(2*n+m,n)*m/(2*n+m)} /* Define this sequence: */ {a(n)=sum(i=0,n-1,(-1)^i*(3+i)!*Stirling2(n,3+i)*Catalan(3,i)/3!)}

a(n) = Sum_{i=0..n-1} (-1)^i*(3+i)!*Stirling2(n,3+i)*Catalan(3,i)/3!, where Stirling2(n,k) = A008277(n,k), Catalan(k,i) = C(2*i+k,i)*k/(2*i+k) = coefficient of x^i in C(x)^k with C(x) = (1-sqrt(1-4x))/(2x).

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

A097558 Sum{k=1 to oo} a(k)/k^r = sqrt(zeta(r) -3/4) +1/2.

Original entry on oeis.org

1, 1, 1, 0, 1, -1, 1, 1, 0, -1, 1, 3, 1, -1, -1, -1, 1, 3, 1, 3, -1, -1, 1, -7, 0, -1, 1, 3, 1, 7, 1, 3, -1, -1, -1, -12, 1, -1, -1, -7, 1, 7, 1, 3, 3, -1, 1, 19, 0, 3, -1, 3, 1, -7, -1, -7, -1, -1, 1, -27, 1, -1, 3, -6, -1, 7, 1, 3, -1, 7, 1, 45, 1, -1, 3, 3, -1, 7, 1, 19, -1, -1, 1, -27, -1, -1, -1, -7, 1, -27, -1, 3, -1, -1, -1, -51, 1, 3, 3, -12, 1, 7

Offset: 1

Leroy Quet, Aug 27 2004

sign

The "+ 1/2" in the Dirichlet series generating function was added so the first term of the sequence is an integer. We could have added/subtracted any other integer+1/2 instead and then had the first term equal another integer. "zeta(r)" refers to sum{k=1 to oo} 1/k^r.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001222, A005043, A048287.

A[1]:= 1:
for n from 2 to 100 do
  A[n]:= 1 - add(A[n/k]*A[k], k= numtheory:-divisors(n) minus {1,n})
od:
seq(A[n],n=1..100); # Robert Israel, Mar 01 2016

a(1)=1; for n>=2, a(n) = 1 - sum{k|n, 2<=k<=n-1} a(n/k) a(k).
From Robert Israel, Mar 01 2016: (Start)
a(n) depends only on the prime signature of n.
If p is prime, a(p^k) = (-1)^(k+1)*A005043(k-1).
If n is squarefree, a(n) = (-1)^(A001222(n)-1)*A048287(A001222(n)).
(End)

More terms from David Wasserman, Dec 27 2007

A136588 a(n) = Sum_{k=0..n} A136595(n,k)*n^k.

Original entry on oeis.org

1, 1, 2, 21, 124, 1880, 20046, 391419, 6195288, 147481299, 3121373690, 87790122816, 2329580861268, 75790954533385, 2415630777959686, 89478235732836705, 3323789119614522416, 138402773923330655700

Offset: 0

Paul D. Hanna, Jan 10 2008

nonn

Cf. A136595; A048287, A136596, A136597; A136589.

{a(n)=sum(k=0,n,if(k==0,0^n,n^k*n!/(k-1)!* sum(i=0,n-1,(-1)^i*polcoeff(((exp(x+x*O(x^n))-1)^(k+i)),n)*binomial(2*i+k,i)/(2*i+k))))}

A136589 a(n) = Sum_{k=0..n} A136595(n,k)*(n+1)^k.

Original entry on oeis.org

1, 2, 6, 44, 345, 4182, 53571, 905144, 16154550, 349976610, 8015743483, 211987872516, 5925745900685, 185188128471374, 6108604964499810, 220403394009702384, 8375181440031684305, 342816825221831030490

Offset: 0

Paul D. Hanna, Jan 10 2008

nonn

Cf. A136595; A048287, A136596, A136597; A136588.

{a(n)=sum(k=0,n,if(k==0,0^n,(n+1)^k*n!/(k-1)!* sum(i=0,n-1,(-1)^i*polcoeff(((exp(x+x*O(x^n))-1)^(k+i)),n)*binomial(2*i+k,i)/(2*i+k))))}

A120903 Triangle T, read by rows, that satisfies matrix equation: T + (T-I)^2 = C, where C is Pascal's triangle.

Original entry on oeis.org

1, 1, 1, -1, 2, 1, 7, -3, 3, 1, -61, 28, -6, 4, 1, 751, -305, 70, -10, 5, 1, -11821, 4506, -915, 140, -15, 6, 1, 226927, -82747, 15771, -2135, 245, -21, 7, 1, -5142061, 1815416, -330988, 42056, -4270, 392, -28, 8, 1, 134341711, -46278549, 8169372, -992964, 94626, -7686, 588, -36, 9, 1, -3975839341, 1343417110

Offset: 0

Paul D. Hanna, Jul 17 2006

Column 0 is signed A048287, which is the number of semiorders on n labeled nodes.

			Triangle T begins:
         1;
         1,       1;
        -1,       2,       1;
         7,      -3,       3,     1;
       -61,      28,      -6,     4,     1;
       751,    -305,      70,   -10,     5,   1;
    -11821,    4506,    -915,   140,   -15,   6,   1;
    226927,  -82747,   15771, -2135,   245, -21,   7, 1;
  -5142061, 1815416, -330988, 42056, -4270, 392, -28, 8, 1;
The matrix square of T less the diagonal is (T-I)^2:
      0;
      0,     0;
      2,     0,   0;
     -6,     6,   0,    0;
     62,   -24,  12,    0,  0;
   -750,   310, -60,   20,  0, 0;
  11822, -4500, 930, -120, 30, 0, 0;
where C = T + (T-I)^2 = 2*T^2 - 2*T + I.

Cf. A048287 (column 0); A117269 (variant: T - (T-I)^2 = C).

/* Generated by Recursion T = C - (T-I)^2 : */ {T(n, k)=local(C=matrix(n+1, n+1, r, c, if(r>=c, binomial(r-1, c-1))), M=C); for(i=1, n+1, M=C-(M-M^0)^2 ); return(M[n+1, k+1])}

/* Generated by E.G.F.: */ {T(n,k)=n!*polcoeff(polcoeff(exp(x*y)*(1 + sqrt(4*exp(x +x*O(x^n))-3))/2,n,x),k,y)}

E.g.f. A(x,y) satisfies: A(x,y) + [A(x,y) - exp(x*y)]^2 = exp(x+x*y).
Explicitly, e.g.f.: A(x,y) = exp(x*y)*(1 + sqrt(4*exp(x)-3))/2.
E.g.f. of column 0: (1 + sqrt(4*exp(x)-3))/2.
T(n,k) = -(-1)^(n-k)*A048287(n-k)*C(n,k) + 2*0^(n-k).
Matrix square: [T^2](n,k) = ( C(n,k) + 2*T(n,k) - 0^(n-k) )/2.

cp's OEIS Frontend

A052851 Expansion of e.g.f. 1/2 - (1/2)*(1+4*log(1-x))^(1/2).

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A136595 Matrix inverse of triangle A136590.

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

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A136596 Column 2 of triangle A136595.

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A136597 Column 3 of triangle A136595.

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

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A097558 Sum{k=1 to oo} a(k)/k^r = sqrt(zeta(r) -3/4) +1/2.

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A136588 a(n) = Sum_{k=0..n} A136595(n,k)*n^k.

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A052851 Expansion of e.g.f. 1/2 - (1/2)(1+4log(1-x))^(1/2).