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A113326 Table T(n,k), n>=1 and k>=0, read by antidiagonals, related to A111146.

%I A113326 #13 Sep 07 2024 21:10:10
%S A113326 1,1,1,1,2,2,1,3,8,5,1,4,18,36,15,1,5,32,117,176,54,1,6,50,272,801,
%T A113326 928,235,1,7,72,525,2400,5724,5296,1237,1,8,98,900,5675,21792,42633,
%U A113326 33024,7790,1,9,128,1421,11520,62650,203008,331911,227776,57581
%N A113326 Table T(n,k), n>=1 and k>=0, read by antidiagonals, related to A111146.
%F A113326 T(n, k) = Sum_{j=0..k} n^j*A111146(k, j).
%F A113326 G.f. for row n: Sum_{k>=0}T(n, k)*y^k = 1/(1-n/(n-1)!*y*Sum_{j>=0}(n-1+j)!*y^j), for n>=1.
%e A113326 Table begins:
%e A113326 1,1,2,5,15,54,235,1237,7790,57581, 489231, ...
%e A113326 1,2,8,36,176,928,5296,33024,227776,1757504, ...
%e A113326 1,3,18,117,801,5724,42633,331911,2717874,23620329, ...
%e A113326 1,4,32,272,2400,21792,203008,1940224,19065344,193410560, ...
%e A113326 1,5,50,525,5675,62650,703975,8042625,93454750,1106250125, ...
%e A113326 1,6,72,900,11520,149904,1976400,26363232,355648320, ...
%e A113326 1,7,98,1421,21021,315168,4774021,72945859,1123559906, ...
%e A113326 1,8,128,2112,35456,601984,10306048,177639936,3080264704, ...
%e A113326 1,9,162,2997,56295,1067742,20392803,391614669,7555447854, ...
%e A113326 1,10,200,4100,85200,1785600,37644400,797224000,16946456000,
%e A113326 ...
%o A113326 (PARI) {T(n,k)=local(y=Y+Y*O(Y^k)); polcoeff(1/(1-n/(n-1)!*y*sum(j=0,k,(n-1+j)!*y^j)),k,Y)}
%Y A113326 Cf. A111146, A051295 (row n=1), A113327 (row n=2), A113328 (row n=3), A113329 (row n=4), A113330 (row n=5), A113331 (row n=6).
%K A113326 nonn,tabl
%O A113326 0,5
%A A113326 _Philippe Deléham_ and _Paul D. Hanna_, Oct 28 2005