A247502 Triangle read by rows: coefficients of polynomials related to the exponential generating function of sequences generated by Narayana polynomials evaluated at the integers; n>=1, 0<=k

1, 1, 1, 1, 4, 1, 1, 13, 9, 1, 1, 41, 57, 16, 1, 1, 131, 320, 165, 25, 1, 1, 428, 1711, 1420, 380, 36, 1, 1, 1429, 8967, 11151, 4620, 756, 49, 1, 1, 4861, 46663, 83202, 49665, 12306, 1358, 64, 1, 1, 16795, 242634, 602407, 495327, 172893, 28476, 2262, 81, 1

Offset: 1

Author

Peter Luschny, Nov 18 2014

Keywords

• nonn
• tabl

Comments

Definition: Let N(n,x) = Sum_{j=0..n-1} x^j*C(n,j)^2*(n-j)/(n*(j+1)) for n>0 and N(0,x) = 1, further let p(n,x) be implicitly defined by N(n,k) = k!*[x^k](exp(x)*p(n,x)), then T(n,k) = [x^k] p(n,x).

Examples

			Triangle T(n,k) begins:
[n\k][0,    1,     2,     3,     4,     5,    6,  8, 9]
[1]   1,
[2]   1,    1,
[3]   1,    4,     1,
[4]   1,   13,     9,     1,
[5]   1,   41,    57,    16,     1,
[6]   1,  131,   320,   165,    25,     1,
[7]   1,  428,  1711,  1420,   380,    36,    1,
[8]   1, 1429,  8967, 11151,  4620,   756,   49,  1,
[9]   1, 4861, 46663, 83202, 49665, 12306, 1358, 64, 1.
.
The sequence N(7,k) = 1 + 21*k + 105*k^2 + 175*k^3 + 105*k^4 + 21*k^5 + k^6 = 1, 429, 4279, 20071, 65445, ... = A090200(k) has the exponential generating function exp(x)*(1 + 428*x + 1711*x^2 + 1420*x^3 + 380*x^4 + 36*x^5 + x^6). Thus T(7,3) = 1420.
		

Crossrefs

Cf. A243631 and the crossreferences given there.

Formula

T(n, 0) = T(n, n-1) = 1.

T(n, 1) = A001453(n) = A000108(n) - 1 for n>=2.

T(n, n-2) = (n-1)^2 for n>=2.