A105556 Triangle, read by rows, such that column n equals the row sums of A001263^n, which is the n-th matrix power of the Narayana triangle A001263, for n>=0.

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, 1, 16796, 448023, 885868, 444225, 99156

Offset: 0

Author

Paul D. Hanna, Apr 14 2005

Keywords

• nonn
• tabl

Comments

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

Examples

			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;
  ...
From _Paul D. Hanna_, 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)
		

Crossrefs

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

Cf. A155926. - Paul D. Hanna, Feb 01 2009

Programs

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)} \\ Paul D. Hanna, Feb 01 2009

Formula

From Paul D. Hanna, 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)