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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

%I A247502 #14 Nov 11 2024 03:57:28
%S A247502 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 A247502 380,36,1,1,1429,8967,11151,4620,756,49,1,1,4861,46663,83202,49665,
%U A247502 12306,1358,64,1,1,16795,242634,602407,495327,172893,28476,2262,81,1
%N 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<n.
%C A247502 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).
%F A247502 T(n, 0) = T(n, n-1) = 1.
%F A247502 T(n, 1) = A001453(n) = A000108(n) - 1 for n>=2.
%F A247502 T(n, n-2) = (n-1)^2 for n>=2.
%e A247502 Triangle T(n,k) begins:
%e A247502 [n\k][0,    1,     2,     3,     4,     5,    6,  8, 9]
%e A247502 [1]   1,
%e A247502 [2]   1,    1,
%e A247502 [3]   1,    4,     1,
%e A247502 [4]   1,   13,     9,     1,
%e A247502 [5]   1,   41,    57,    16,     1,
%e A247502 [6]   1,  131,   320,   165,    25,     1,
%e A247502 [7]   1,  428,  1711,  1420,   380,    36,    1,
%e A247502 [8]   1, 1429,  8967, 11151,  4620,   756,   49,  1,
%e A247502 [9]   1, 4861, 46663, 83202, 49665, 12306, 1358, 64, 1.
%e A247502 .
%e A247502 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 A247502 Cf. A243631 and the crossreferences given there.
%K A247502 nonn,tabl
%O A247502 1,5
%A A247502 _Peter Luschny_, Nov 18 2014