A197927 -id:A197927 - cp's OEIS frontend

A202947 G.f.: [ Sum_{n>=0} (n+1) * 2^(n^2) * x^n ]^(1/2).

1, 2, 22, 980, 161638, 100318460, 240313495420, 2251316821283048, 83005840299778004614, 12089092134684999622076396, 6972054121242613685463168904468, 15950722005044706228925521886595357720, 144954811888851643278920459489891540357638876

Offset: 0

Paul D. Hanna, Dec 26 2011

nonn

Equals the self-convolution square-root of A197927 (with offset).

			G.f.: A(x) = 1 + 2*x + 22*x^2 + 980*x^3 + 161638*x^4 + 100318460*x^5 +...
where
A(x)^2 = 1 + 2*2*x + 3*2^4*x^2 + 4*2^9*x^3 + 5*2^16*x^4 + 6*2^25*x^5 +...
more explicitly,
A(x)^2 = 1 + 4*x + 48*x^2 + 2048*x^3 + 327680*x^4 + 201326592*x^5 +...+ A197927(n+1)*x^n +...

Cf. A197927, A202942.

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

{a(n)=if(n==0,1,(n+1)*2^(n^2-1)-sum(k=1,n-1,a(n-k)*a(k)/2))}

a(n) = (n+1)*2^(n^2-1) - Sum_{k=1..n-1} a(n-k)*a(k)/2 for n>0 with a(0)=1.

A217436 Triangular array read by rows. T(n,k) is the number of labeled relations on n elements with exactly k vertices of indegree and outdegree = 0.

Original entry on oeis.org

1, 1, 1, 13, 2, 1, 469, 39, 3, 1, 63577, 1876, 78, 4, 1, 33231721, 317885, 4690, 130, 5, 1, 68519123173, 199390326, 953655, 9380, 195, 6, 1, 562469619451069, 479633862211, 697866141, 2225195, 16415, 273, 7, 1, 18442242396353040817, 4499756955608552, 1918535448844, 1860976376, 4450390, 26264, 364, 8, 1

Offset: 0

Geoffrey Critzer, Oct 02 2012

Row sums = 2^(n^2). First column (k = 0) is A173403.

Sum_{k=1,2,...,n} T(n,k)*k = A197927.

			1,
1, 1,
13, 2, 1,
469, 39, 3, 1,
63577, 1876, 78, 4, 1,
33231721, 317885, 4690, 130, 5, 1,
68519123173, 199390326, 953655, 9380, 195, 6, 1

nn=6; s=Sum[Sum[(-1)^k Binomial[n,k] 2^(n-k)^2, {k,0,n}] x^n/n!, {n,0,nn}]; Range[0,nn]! CoefficientList[Series[Exp[ y x] s, {x,0,nn}], {x,y}] //Grid

E.g.f.: exp(y*x)*A(x) where A(x) is the e.g.f. for A173403.

cp's OEIS Frontend

A202947 G.f.: [ Sum_{n>=0} (n+1) * 2^(n^2) * x^n ]^(1/2).

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