A273895 T(n, k) is the number of Horizontal Convex Polyominoes with n cells and k rows.

0, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 9, 8, 1, 0, 1, 16, 31, 12, 1, 0, 1, 25, 85, 68, 16, 1, 0, 1, 36, 190, 260, 121, 20, 1, 0, 1, 49, 371, 777, 604, 190, 24, 1, 0, 1, 64, 658, 1960, 2299, 1180, 275, 28, 1, 0, 1, 81, 1086, 4368, 7221, 5509, 2052, 376, 32, 1, 0

Offset: 0

Author

Michael Somos, Jun 02 2016

Keywords

• nonn

Examples

			Triangle begins:
0,
0, 1,
0, 1, 1,
0, 1, 4, 1,
0, 1, 9, 8, 1,
		

Links

• R. Pemantle and M. C. Wilson, Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions, SIAM Rev., 50 (2008), no. 2, 199-272. See p. 239

Crossrefs

Cf. A001169.

Programs

Mathematica

T[n_, m_] := Sum[Sum[Sum[Binomial[i - 2*j, j]*2^(i - 3*j)*Binomial[k + j, i - 2*j]*Binomial[k + 3*j - i, m + j - i - 1], {j, 0, m + i - 1}]*Binomial[ n - k - 2, n - k - i - 1], {i, 0, n - k - 1}], {k, 0, n - 1}]; Table[T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jan 27 2019, after Vladimir Kruchinin *)

Maxima

T(n,m):=sum(sum((sum(binomial(i-2*j,j)*2^(i-3*j)*binomial(k+j,i-2*j)*binomial(k+3*j-i,m+j-i-1),j,0,m+i-1))*binomial(n-k-2,n-k-i-1),i,0,n-k-1),k,0,n-1); /* Vladimir Kruchinin, Jan 27 2019 */

PARI

{T(n, k) = if( k<0 || k>n, 0, polcoeff( polcoeff( x * y *(1 - x)^3 / ((1 - x)^4 - x * y * (1 - x - x^2 + x^3 + x^2 * y)) + x * O(x^n), n), k))};

Formula

G.f.: x * y * (1 - x)^3 / ((1 - x)^4 - x * y * (1 - x - x^2 + x^3 + x^2 * y)) = Sum_{0<=k<=n} T(n, k) * x^n * y^k.

Row sums are A001169.

T(n,m) = Sum_{k=0..n-1} Sum_{i=0..n-k-1} [Sum_{j=0..m+i-1} C(i-2*j,j)*2^(i-3*j)*C(k+j,i-2*j)*C(k+3*j-i,m+j-i-1)]*C(n-k-2,n-k-i-1). - Vladimir Kruchinin, Jan 27 2019