A103370 -id:A103370 - cp's OEIS frontend

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

Paul D. Hanna, Apr 14 2005

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

			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)

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

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

{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])}

{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

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)

A112338 Triangle read by rows, generated from A001263.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 12, 14, 1, 1, 5, 22, 57, 42, 1, 1, 6, 35, 148, 303, 132, 1, 1, 7, 51, 305, 1144, 1743, 429, 1, 8, 70, 546, 3105, 9784, 10629, 1430, 1

Offset: 0

Gary W. Adamson, Sep 04 2005

Rows of the array are row sums of n-th powers of the Narayana triangle; e.g., row 1 = A000108: (1, 2, 5, 14, 42, ...); row 2 = row sums of the Narayana triangle squared (A103370): (1, 3, 12, 57, 303, ...), etc.

			In the array, antidiagonal terms (1, 3, 5, 1) become row 3 of the triangle.
First few rows of the array:
  1, 1,  1,   1,    1,     1, ...
  1, 2,  5,  14,   42,   132, ...
  1, 3, 12,  57,  303,  1743, ...
  1, 4, 22, 148, 1144,  9784, ...
  1, 5, 35, 305, 3105, 35505, ...
First few rows of the triangle:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  5,   1;
  1, 4, 12,  14,   1;
  1, 5, 22,  57,  42,   1;
  1, 6, 35, 148, 303, 132, 1;

Cf. A001263, A000326, A005915, A095266, A000108, A103370.

Let M be the infinite lower triangular Narayana triangle (A001263). Perform M^n * [1 0 0 0 ...] getting an array. Take antidiagonals of the array which become rows of the triangle A112338.

cp's OEIS Frontend

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.

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