0,5

Column 1 is the Catalan numbers A000108 (offset 1).

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Feb 01 2009: (Start)

G.f. of column k = B(x)^(k+1) where B(x) = Sum_{n>=0} x^n/[n!*(n+1)!/2^n];

T(n,K) = [x^(n-k)] B(x)^(k+1) * (n-k)!*(n-k+1)!/2^(n-k) for n>=k>=0. (End)

Triangle begins:

1;

1,1;

1,2,1;

1,5,3,1;

1,14,12,4,1;

1,42,57,22,5,1;

1,132,303,148,35,6,1;

1,429,1743,1144,305,51,7,1;

1,1430,10629,9784,3105,546,70,8,1;

1,4862,67791,90346,35505,6906,889,92,9,1; ...

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Feb 01 2009: (Start)

G.f. for rows n=0..3 are:

B(x) = 1 + x + x^2/3 + x^3/18 + x^4/180 + x^5/2700 +...+ x^n/[n!*(n+1)!/2^n] +...

B(x)^2 = 1 + 2*x + 5*x^2/3 + 14*x^3/18 + 42*x^4/180 +...+ A000108(n)*x^n/[n!*(n+1)!/2^n] +...

B(x)^3 = 1 + 3*x +12*x^2/3 + 57*x^3/18 +303*x^4/180 +...+ A103370(n)*x^n/[n!*(n+1)!/2^n] +...

B(x)^4 = 1 + 4*x +22*x^2/3 +148*x^3/18+1144*x^4/180 + 9784*x^5/2700 + 90346*x^5/56700 +... (End)

(PARI) {T(n, k)=local(N=matrix(n+1, n+1, m, j, if(m>=j, binomial(m-1, j-1)*binomial(m, j-1)/j))); sum(j=0, n-k, (N^k)[n-k+1, j+1])}

(PARI) {T(n, k)=local(B=sum(j=0, n-k, x^j/(j!*(j+1)!/2^j))+x*O(x^(n-k))); polcoeff(B^(k+1), n-k)*(n-k)!*(n-k+1)!/2^(n-k)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Feb 01 2009]

Cf. A001263, A105557 (row sums), A103370 (column 2).

Cf. A155926. [From Paul D. Hanna (pauldhanna(AT)juno.com), Feb 01 2009]

Sequence in context: A106240 A097615 A062993 this_sequence A078920 A117396 A125860

Adjacent sequences: A105553 A105554 A105555 this_sequence A105557 A105558 A105559

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Apr 14 2005