A243950 -id:A243950 - cp's OEIS frontend

A243951 Self-convolution square-root of A243950, which is the sums of the squares of the q-binomial coefficients for q=2 in rows of triangle A022166.

1, 1, 5, 45, 781, 23981, 1371885, 145101805, 29560055405, 11546945197165, 8881721878376045, 13338290506465706605, 39879639563413780322925, 234862804790553590007179885, 2768979430068663216466330446445, 64586918396493458414460474344516205, 3024204274887062319005574660727125346925

Offset: 0

Paul D. Hanna, Jun 21 2014

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 45*x^3 + 781*x^4 + 23981*x^5 + 1371885*x^6 +...
where
A(x)^2 = 1 + 2*x + 11*x^2 + 100*x^3 + 1677*x^4 + 49974*x^5 + 2801567*x^6 + 293257480*x^7 + 59426801521*x^8 +...+ A243950(n)*x^n +...
The terms in this sequence appear to be divisible by 5 everywhere except
a(n) == 1 (mod 5) when n = {0,1,4,5,20,21,24,25,100,101,104,105,120,121,124, 125,500,501,...}.

Paul D. Hanna, Table of n, a(n) for n = 0..80

Cf. A243950, A022166.

a[n_] := SeriesCoefficient[Sqrt[Sum[x^m Sum[QBinomial[m, k, 2]^2, {k, 0, m}], {m, 0, n}]], {x, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Apr 09 2016 *)

{A022166(n, k)=polcoeff(x^k/prod(j=0, k, 1-2^j*x+x*O(x^n)), n)}
{a(n)=polcoeff(sqrt(sum(m=0,n,x^m*sum(k=0,m,A022166(m, k)^2) +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))

a(n) ~ c * 2^(n^2/2-1), where c = 18.0796893855819714431... if n is even and c = 18.02126069886312898683... if n is odd (constants same as for A243950). - Vaclav Kotesovec, Jun 23 2014

A359313 Triangular array read by rows. T(n,k) is the number of Green's H-classes contained in the D-class of rank k matrices in the semigroup Mat_n(F_2) of n X n matrices over the field F_2. n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 49, 49, 1, 1, 225, 1225, 225, 1, 1, 961, 24025, 24025, 961, 1, 1, 3969, 423801, 1946025, 423801, 3969, 1, 1, 16129, 7112889, 139499721, 139499721, 7112889, 16129, 1, 1, 65025, 116532025, 9439094025, 40315419369, 9439094025, 116532025, 65025, 1

Offset: 0

Geoffrey Critzer, Dec 25 2022

For all a,b in the semigroup Mat_n(F_2), aDb if and only if rank(a)=rank(b). Also, aHb if and only if the row(a)=row(b) and col(a)=col(b). So the H classes correspond to ordered pairs (U,W) of subspaces of F_2^n with dim(U) = dim(W). Let a in Mat_n(F_2) be such that col(a) = U and row(a)=W. The size of H_a, the H-class containing a is |GL_d(F_2)| where d=dim(U). H_a contains an idempotent if and only if col(a) + perp(row(a)) is a direct sum decomposition of F_2^n where perp(X)={v in F_2^n: v*x = 0 for all x in X}.

Let H_a,H_b be H-classes in Mat_n(F_2). Let H_a ~ H_b if and only if col(a) is contained in col(b) and row(a) is contained in row(b). Then ~ is a partial order relation on the set of all H-classes in Mat_n(F_q). The poset is isomorphic to a typical n-interval in the binomial poset L*L where L is the binomial poset of all finite dimensional subspaces over a countably infinite dimensional subspace and * is the Segre product (see Stanley reference). T(n,k) is the number of elements at rank k in an n-interval of L*L.

			 1
 1,   1
 1,   9,     1
 1,  49,    49,     1
 1, 225,  1225,   225,   1
 1, 961, 24025, 24025, 961, 1

R. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, section 3.18.

Wikipedia, Green's relations

Cf. A243950 (row sums), A022166, A005329, A002884, A002416, A296548 (a subposet of L*L).

nn = 8; B[n_, q_] := QFactorial[n, q]^2; e[x_] := Sum[x^n/B[n, 2], {n, 0, nn}]; Map[Select[#, # > 0 &] &, Table[QFactorial[n, 2]^2, {n, 0, nn}] CoefficientList[
   Series[e[x] e[y x], {x, 0, nn}], {x, y}]]

T(n,k) = A022166(n,k)^2.
Sum_{k=0..n} T(n,k)*A002884(k) = A002416(n).
Let B(n) = A005329(n)^2.  Let E(x)=Sum_{n>=0}x^n/B(n).  Then Sum{n>=0} Sum{k=0..n} T(n,k)*y^k*x^n/B(n) = E(x)*E(y*x).

cp's OEIS Frontend

A243951 Self-convolution square-root of A243950, which is the sums of the squares of the q-binomial coefficients for q=2 in rows of triangle A022166.

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