A101370 -id:A101370 - cp's OEIS frontend

A167139 G.f.: Sum_{n>=0} A005649(n)^2 * log(1+x)^n/n! where 1/(1-x)^2 = Sum_{n>=0} A005649(n)*log(1+x)^n/n!.

1, 4, 30, 292, 3497, 49488, 806504, 14860032, 305261640, 6914828176, 171186477632, 4597513706496, 133116705145408, 4133143450593536, 136981118139314688, 4826352390162440704, 180139085757269111824

Offset: 0

Paul D. Hanna, Nov 03 2009

nonn

Conjecture: For all integers m > 0, Sum_{n>=0} L(n)^m * log(1+x)^n/n! is an integer series whenever Sum_{n>=0} L(n)*log(1+x)^n/n! is an integer series.

			G.f.: A(x) = 1 + 4*x + 30*x^2 + 292*x^3 + 3497*x^4 + 49488*x^5 + ...
Illustrate A(x) = Sum_{n>=0} A005649(n)^2 * log(1+x)^n/n!:
A(x) = 1 + 2^2*log(1+x) + 8^2*log(1+x)^2/2! + 44^2*log(1+x)^3/3! + 308^2*log(1+x)^4/4! + 2612^2*log(1+x)^5/5! + ... + A005649(n)^2*log(1+x)^n/n! + ...
where the g.f. of A005649 is 1/(2 - exp(x))^2:
1/(1-x)^2 = 1 + 2*log(1+x) + 8*log(1+x)^2/2! + 44*log(1+x)^3/3! + 308*log(1+x)^4/4! + 2612*log(1+x)^5/5! + ... + A005649(n)*log(1+x)^n/n! + ...

Cf. A167138, A005649.

{A005649(n)=sum(k=0,n,(k+1)*stirling(n, k, 2)*k!)}
{a(n)=polcoef(sum(m=0,n,A005649(m)^2*log(1+x+x*O(x^n))^m/m!),n)}

a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A005649(k)^2, cf. A101370. - Vladeta Jovovic, Nov 09 2009

A365961 Number of (0,1)-matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums.

Original entry on oeis.org

1, 1, 4, 19, 127, 967, 9063, 94595, 1139708, 15118010, 223571836, 3597458356, 63233950081, 1197193320701, 24418765771835, 532015160784016, 12363381055074017, 304754656068754421, 7952728315095555279, 218848562411197549582, 6338152295627215890669, 192627799720153909693048

Offset: 0

Ludovic Schwob, Sep 23 2023

nonn

Let f(n) = number of ordered coprime factorizations of n (A325446(n)); a(n) = sum of f(k) over all terms k in A025487 that have n factors.

			The a(3) = 19 matrices:
  [1 1 1]
.
  [1 1] [1 1] [1 1 0] [1 0 1] [0 1 1]
  [1 0] [0 1] [0 0 1] [0 1 0] [1 0 0]
.
  [1] [1 0] [0 1] [1 0] [0 1] [1 0 0] [1 0 0] [0 1] [1 0]
  [1] [1 0] [0 1] [0 1] [1 0] [0 1 0] [0 0 1] [1 0] [0 1]
  [1] [0 1] [1 0] [1 0] [0 1] [0 0 1] [0 1 0] [1 0] [0 1]
.
  [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [1 0 0] [0 1 0] [1 0 0]

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A068313, A035341, A321735, A101370, A321733, A104602, A116540.

R(n,k)={Vec(-1 + 1/prod(j=1, k, 1 - binomial(k,j)*x^j + O(x*x^n)))}
seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Sep 23 2023

Terms a(13) and beyond from Andrew Howroyd, Sep 23 2023

A122725 a(n) = A000670(n)^2.

Original entry on oeis.org

1, 1, 9, 169, 5625, 292681, 21930489, 2236627849, 297935847225, 50229268482121, 10454564139438969, 2632936466960600329, 789136169944454084025, 277579719258755165321161, 113238180214596650771616249, 53030348046942317338336489609, 28256184698070300360908567636025

Offset: 0

Vladeta Jovovic, Sep 23 2006

This is also the number of possible positions of n intervals on a line having a common non-punctual intersection. Proof: Let us denoted each interval Ai (1 <= i <= n) by the string AiAi. Then the set of all such relative positions is given by the S-language [A1 ⊗ A2 ... ⊗ An]^2. The cardinality of $A1 ⊗ A2 ... ⊗ An$ is given by A000670. - Sylviane R. Schwer (schwer(AT)lipn.univ-paris13.fr), Nov 26 2007

Cf. A000670, A055203, A101370.

b:= proc(n, k) option remember;
     `if`(n=0, 1, add(b(n-1, j)*j, j=k..k+1))
    end:
a:= n-> b(n, 0)^2:
seq(a(n), n=0..16);  # Alois P. Heinz, Aug 12 2025

Table[(PolyLog[ -z, 1/2]/2)^2, {z, 1, 16}] (* Elizabeth A. Blickley (Elizabeth.Blickley(AT)gmail.com), Oct 10 2006 *)

{a(n)=sum(k=0, n, stirling(n, k, 2)*k!)^2} \\ Paul D. Hanna, Nov 07 2009

a(n) = Sum_{m>=0} Sum_{k>=0} ((k*m)^n/2^(k+m+2)).
G.f.: Sum_{n>=0} (1/(2-exp(n*x))/2^(n+1)).
Sum_{n>=0} a(n)*log(1+x)^n/n! = o.g.f. of A101370. - Paul D. Hanna, Nov 07 2009
a(n) ~ (n!)^2 / (4 * (log(2))^(2*n+2)). - Vaclav Kotesovec, May 03 2015

More terms from Elizabeth A. Blickley (Elizabeth.Blickley(AT)gmail.com), Oct 10 2006

A321584 Number of connected (0,1)-matrices with n ones and no zero rows or columns.

Original entry on oeis.org

1, 1, 2, 6, 27, 159, 1154, 9968, 99861, 1138234, 14544650, 205927012, 3199714508, 54131864317, 990455375968, 19488387266842, 410328328297512, 9205128127109576, 219191041679766542, 5521387415218119528, 146689867860276432637, 4099255234885039058842, 120199458455807733040338

Offset: 0

Gus Wiseman, Nov 13 2018

nonn

A matrix is connected if the positions in each row (or each column) of the nonzero entries form a connected hypergraph.

			The a(4) = 27 matrices:
  [1111]
.
  [111][111][111][11][110][110][101][101][100][011][011][010][001]
  [100][010][001][11][101][011][110][011][111][110][101][111][111]
.
  [11][11][11][11][10][10][10][10][01][01][01][01]
  [10][10][01][01][11][11][10][01][11][11][10][01]
  [10][01][10][01][10][01][11][11][10][01][11][11]
.
  [1]
  [1]
  [1]
  [1]

Andrew Howroyd, Table of n, a(n) for n = 0..100

Cf. A007716, A007718, A049311, A056156, A101370, A104602, A120733, A283877, A319557, A319647, A319616-A319629, A321585.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],Length[csm[Map[Last,GatherBy[#,First],{2}]]]==1]&]],{n,6}] (* Mathematica 7.0+ *)

NonZeroCols(M)={my(C=Vec(M)); Mat(vector(#C, n, sum(k=1, n, (-1)^(n-k)*binomial(n,k)*C[k])))}
ConnectedMats(M)={my([m,n]=matsize(M), R=matrix(m,n)); for(m=1, m, for(n=1, n, R[m,n] = M[m,n] - sum(i=1, m-1, sum(j=1, n-1, binomial(m-1,i-1)*binomial(n,j)*R[i,j]*M[m-i,n-j])))); R}
seq(n)={my(M=matrix(n,n,i,j,sum(k=1, n, binomial(i*j,k)*x^k, O(x*x^n) ))); Vec(1 + vecsum(vecsum(Vec( ConnectedMats( NonZeroCols( NonZeroCols(M)~)) ))))} \\ Andrew Howroyd, Jan 17 2024

a(7) onwards from Andrew Howroyd, Jan 17 2024

A123114 a(n) = Sum_{r>0,s>0} binomial(r*s-1,n-1)/2^(r+s).

Original entry on oeis.org

1, 3, 13, 83, 701, 7363, 92541, 1354627, 22636861, 425241347, 8871085565, 203487078403, 5090418231549, 137920771272963, 4023549748488445, 125743894742698243, 4191213031967650813, 148414827031140706307

Offset: 1

Vladeta Jovovic, Sep 28 2006

Vaclav Kotesovec, Table of n, a(n) for n = 1..400
M. Maia and M. Mendez, On the arithmetic product of combinatorial species

Cf. A101370, A000629.

Table[Sum[StirlingS1[n, k]*(Sum[(j - 1)!*StirlingS2[k, j], {j, 1, k}])^2, {k, 1, n}]/(n-1)!, {n, 1, 20}] (* Vaclav Kotesovec, Jun 07 2019 *)
Table[-(-1)^n + Sum[StirlingS1[n, k]*PolyLog[1-k, 2]^2, {k, 2, n}]/(n-1)!, {n, 1, 20}] (* Vaclav Kotesovec, Jun 07 2019 *)

a(n) = (1/(n-1)!)*Sum_{k=1..n} Stirling1(n,k)*b(k)^2, where b(n) = Sum_{k=1..n} (k-1)!*Stirling2(n,k).

a(n) ~ c * (n-1)! / (log(2))^(2*n), where c = 2^(-log(2)/2) = 0.7864497045594053649114085152934509198700275589579678941719548714254307448... - Vaclav Kotesovec, Jun 07 2019

A167141 G.f.: Sum_{n>=0} A004123(n)^2log(1+x)^n/n! where 1/(1-2x) = Sum_{n>=0} A004123(n)log(1+x)^n/n!.

Original entry on oeis.org

1, 4, 48, 864, 20880, 632448, 23018688, 978179328, 47529084096, 2598928566336, 157937795847936, 10559489876375040, 770269715428025088, 60876094422772800000, 5181654464327251948032, 472584847824904789910016

Offset: 0

Paul D. Hanna, Nov 04 2009

nonn

CONJECTURE: For all integer m>0, Sum_{n>=0} L(n)^m * log(1+x)^n/n! is an integer series whenever Sum_{n>=0} L(n)*log(1+x)^n/n! is an integer series.

In this case, m=2 and L(n) = A004123(n), which is the number of generalized weak orders on n points.

			G.f.: A(x) = 1 + 4*x + 48*x^2 + 864*x^3 + 20880*x^4 + 632448*x^5 +...
Illustrate A(x) = Sum_{n>=0} A004123(n)^2 * log(1+x)^n/n!:
A(x) = 1 + 2^2*log(1+x) + 10^2*log(1+x)^2/2! + 74^2*log(1+x)^3/3! + 730^2*log(1+x)^4/4! + 9002^2*log(1+x)^5/5! +...+ A004123(n)^2*log(1+x)^n/n! +...
where the e.g.f. of A004123 is 1/(3 - 2*exp(x)) and thus:
1/(1-2x) = 1 + 2*log(1+x) + 10*log(1+x)^2/2! + 74*log(1+x)^3/3! + 730*log(1+x)^4/4! + 9002*log(1+x)^5/5! +...+ A004123(n)*log(1+x)^n/n! +...

Cf. A004123, variants: A167139, A167138, A101370.

{A004123(n)=sum(k=0,n,2^k*stirling(n, k, 2)*k!)}
{a(n)=polcoeff(sum(m=0,n,A004123(m)^2*log(1+x+x*O(x^n))^m/m!),n)}

A167532 G.f.: Sum_{n>=0} A155585(n)^2 * log(1/(1-2x))^n/n!, where 1/(1-2x+2x^2) = Sum_{n>=0} A155585(n)log(1/(1-2*x))^n/n!.

Original entry on oeis.org

1, 2, 2, 8, 20, 112, 432, 3200, 16704, 154688, 1017920, 11333888, 90011264, 1172330496, 10908526592, 162802835456, 1737036006400, 29235365490688, 351847501606912, 6593866787569664, 88364197074231296, 1825016315965767680

Offset: 0

Paul D. Hanna, Nov 05 2009

nonn

Note that A155585(n) = 2^n E_{n}(1) where E_{n}(x) are the Euler polynomials; e.g.f. of A155585 is exp(x)/cosh(x).

CONJECTURE: For all integer m>0, Sum_{n>=0} L(n)^m * log(1+x)^n/n! is an integer series whenever Sum_{n>=0} L(n)*log(1+x)^n/n! is an integer series.

			G.f.: A(x) = 1 + 2*x + 2*x^2 + 8*x^3 + 20*x^4 + 112*x^5 + 432*x^6 +...
Illustrate A(x) = Sum_{n>=0} A155585(n)^2*log(1/(1-2*x))^n/n!:
A(x) = 1 - log(1-2*x) - 2^2*log(1-2*x)^3/3! - 16^2*log(1-2*x)^5/5! - 272^2*log(1-2*x)^7/7! - 7936^2*log(1-2*x)^9/9! +...+ A155585(n)^2*[ -log(1-2x)]^n/n! +...
where:
1/((1-x)^2 + x^2) = 1 - log(1-2*x) + 2*log(1-2*x)^3/3! - 16*log(1-2*x)^5/5! + 272*log(1-2*x)^7/7! - 7936*log(1-2*x)^9/9! +...+ A155585(n)*[ -log(1-2x)]^n/n! +...

Cf. A155585, variants: A167141, A167139, A167138, A101370.

{A155585(n)=if(n==0,1,bernfrac(n+1)*(2^(n+1)-1)*2^(n+1)/(n+1))}
{a(n)=polcoeff(sum(k=0,n,A155585(k)^2*log(1/(1-2*x +x*O(x^n)))^k/k!),n)}

cp's OEIS Frontend

A167139 G.f.: Sum_{n>=0} A005649(n)^2 * log(1+x)^n/n! where 1/(1-x)^2 = Sum_{n>=0} A005649(n)*log(1+x)^n/n!.

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A365961 Number of (0,1)-matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums.

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A122725 a(n) = A000670(n)^2.

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A321584 Number of connected (0,1)-matrices with n ones and no zero rows or columns.

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Mathematica

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A123114 a(n) = Sum_{r>0,s>0} binomial(r*s-1,n-1)/2^(r+s).

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A167141 G.f.: Sum_{n>=0} A004123(n)^2*log(1+x)^n/n! where 1/(1-2x) = Sum_{n>=0} A004123(n)*log(1+x)^n/n!.

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A167532 G.f.: Sum_{n>=0} A155585(n)^2 * log(1/(1-2*x))^n/n!, where 1/(1-2*x+2*x^2) = Sum_{n>=0} A155585(n)*log(1/(1-2*x))^n/n!.

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A167141 G.f.: Sum_{n>=0} A004123(n)^2log(1+x)^n/n! where 1/(1-2x) = Sum_{n>=0} A004123(n)log(1+x)^n/n!.

A167532 G.f.: Sum_{n>=0} A155585(n)^2 * log(1/(1-2x))^n/n!, where 1/(1-2x+2x^2) = Sum_{n>=0} A155585(n)log(1/(1-2*x))^n/n!.