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%I #9 Jan 19 2015 06:50:06
%S 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,
%T 380,36,1,1,1429,8967,11151,4620,756,49,1,1,4861,46663,83202,49665,
%U 12306,1358,64,1,1,16795,242634,602407,495327,172893,28476,2262,81,1
%N 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<n.
%C Definition: Let N(n,x) = sum(0<=j<=n-1, x^j*C(n,j)^2*(n-j)/(n*(j+1))
%C 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).
%F T(n, 0) = T(n, n-1) = 1.
%F T(n, 1) = A001453(n) = A000108(n) - 1 for n>=2.
%F T(n, n-2) = (n-1)^2 for n>=2.
%e Triangle T(n,k) begins:
%e [n\k][0, 1, 2, 3, 4, 5, 6, 8, 9]
%e [1] 1,
%e [2] 1, 1,
%e [3] 1, 4, 1,
%e [4] 1, 13, 9, 1,
%e [5] 1, 41, 57, 16, 1,
%e [6] 1, 131, 320, 165, 25, 1,
%e [7] 1, 428, 1711, 1420, 380, 36, 1,
%e [8] 1, 1429, 8967, 11151, 4620, 756, 49, 1,
%e [9] 1, 4861, 46663, 83202, 49665, 12306, 1358, 64, 1.
%e .
%e 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.
%Y Cf. A243631 and the crossreferences given there.
%K nonn,tabl
%O 1,5
%A _Peter Luschny_, Nov 18 2014
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