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%I #12 May 26 2018 22:31:26
%S 1,0,1,2,2,2,15,30,18,6,232,492,400,144,24,5335,12450,11450,5240,1200,
%T 120,175416,439698,447744,237612,69672,10800,720,7847665,20851502,
%U 22993348,13653304,4724328,956760,105840,5040,460083056,1283257192,1509767920,979072928,383250880,92961216,13700928,1128960,40320,34295632587,99690153120,123801126966,86244590412,37181530008,10307520792,1842700968,205630272,13063680,362880,3171572232860,9554655641770,12424884790780,9188854250660,4284505197600,1318457696960,271837330560,37189193040,3241998720,163296000,3628800
%N O.g.f. A(x,y) satisfies: [x^n] exp( n * x*A(x,y) ) * (n + y - A(x,y)) = 0 for n > 0.
%C Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:
%C (1) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0,
%C (2) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0.
%C It is remarkable that this triangle should consist entirely of integers.
%H Paul D. Hanna, <a href="/A305109/b305109.txt">Table of n, a(n) for n = 0..1325 of rows 0..50 of this triangle in flattened form.</a>
%e G.f.: A(x,y) = 1 + x*y + x^2*(2 + 2*y + 2*y^2) + x^3*(15 + 30*y + 18*y^2 + 6*y^3) + x^4*(232 + 492*y + 400*y^2 + 144*y^3 + 24*y^4) + x^5*(5335 + 12450*y + 11450*y^2 + 5240*y^3 + 1200*y^4 + 120*y^5) + x^6*(175416 + 439698*y + 447744*y^2 + 237612*y^3 + 69672*y^4 + 10800*y^5 + 720*y^6) + x^7*(7847665 + 20851502*y + 22993348*y^2 + 13653304*y^3 + 4724328*y^4 + 956760*y^5 + 105840*y^6 + 5040*y^7) + x^8*(460083056 + 1283257192*y + 1509767920*y^2 + 979072928*y^3 + 383250880*y^4 + 92961216*y^5 + 13700928*y^6 + 1128960*y^7 + 40320*y^8) + ...
%e where A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^n*y^k satisfies:
%e [x^n] exp( n * x*A(x,y) ) * (n + y - A(x,y)) = 0 for n > 0.
%e This triangle of coefficients T(n,k) of x^n*y^k in A(x,y) begins:
%e 1;
%e 0, 1;
%e 2, 2, 2;
%e 15, 30, 18, 6;
%e 232, 492, 400, 144, 24;
%e 5335, 12450, 11450, 5240, 1200, 120;
%e 175416, 439698, 447744, 237612, 69672, 10800, 720;
%e 7847665, 20851502, 22993348, 13653304, 4724328, 956760, 105840, 5040;
%e 460083056, 1283257192, 1509767920, 979072928, 383250880, 92961216, 13700928, 1128960, 40320;
%e 34295632587, 99690153120, 123801126966, 86244590412, 37181530008, 10307520792, 1842700968, 205630272, 13063680, 362880; ...
%e in which the main diagonal equals the factorials.
%o (PARI) {T(n,k) = my(A=[1],m); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( exp( (m-1)*x*(Ser(A)) ) * (m-1 + y - Ser(A)) )[m] ); polcoeff(A[n+1],k)}
%o for(n=0,10,for(k=0,n, print1(T(n,k),", "));print(""))
%Y Cf. A305110 (row sums), A305111 (column 0), A305112 (y=2), A305113 (y=3).
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, May 26 2018