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A190015 Triangle T(n,k) for solving differential equation A'(x)=G(A(x)), G(0)!=0.

%I A190015 #31 Feb 13 2016 06:29:54
%S A190015 1,1,2,1,6,8,1,24,42,16,22,1,120,264,180,192,136,52,1,720,1920,1248,
%T A190015 540,1824,2304,272,732,720,114,1,5040,15840,10080,8064,18720,22752,
%U A190015 9612,7056,10224,17928,3968,2538,3072,240,1,40320,146160,92160,70560,32256,207360,249120,193536,73728,61560,144720,246816,101844,142704,7936,51048,110448,34304,8334,11616,494,1
%N A190015 Triangle T(n,k) for solving differential equation A'(x)=G(A(x)), G(0)!=0.
%C A190015 For solving the differential equation A'(x)=G(A(x)), where G(0)!=0,
%C A190015 a(n) = 1/n!*sum(pi(i) in P(2*n-1,n), T(n,i)*prod(j=1..n, g(k_j-1))),
%C A190015 where pi(i) is the partition of 2*n-1 into n parts in lexicographic order P(2*n-1,n).
%C A190015 G(x) = g(0)+g(1)*x+g(2)*x^2+...
%C A190015 Examples
%C A190015 A003422  A'(x)=A(x)+1/(1-x)
%C A190015 A000108  A'(x)=1/(1-2*A(x)),
%C A190015 A001147  A'(x)=1/(1-A(x))
%C A190015 A007489  A'(x)=A(x)+x/(1-x)^2+1.
%C A190015 A006351  B'(x)=(1+B(x))/(1-B(x))
%C A190015 A029768  A'(x)=log(1/(1-A(x)))+1.
%C A190015 A001662  B'(x)=1/(1+B(x))
%C A190015 A180254  A'(x)=(1-sqrt(1-4*A(x)))/2
%C A190015 Compare with A145271. There (j')^k = [(d/dx)^j g(x)]^k evaluated at x=0 gives formulas expressed in terms of the coefficients of the Taylor series g(x). If, instead, we express the formulas in terms of the coefficients of the power series of g(x), we obtain a row reversed array for A190015 since the partitions there are in reverse order to the ones here. Simply exchange (j!)^k * (j")^k for (j')^k, where (j")^k = [(d/dx)^j g(x) / j!]^k, to transform from one array to the other. E.g., R^3 g(x) = 1 (0')^1 (1')^3 + 4 (0')^2 (1')^1 (2')^1 + 1 (0')^3 (3')^1 = 1 (O")^1 (1")^3 + 4 (0")^2 (1")^1 2*(2")^1 + 1 (0")^1 3!*(3")^1 = 1 (O")^1 (1")^3 + 8 (0")^2 (1")^1 (2")^1 + 6 (0")^1 (3")^1, the fourth partition polynomial here. - _Tom Copeland_, Oct 17 2014
%e A190015 Triangle begins:
%e A190015 1;
%e A190015 1;
%e A190015 2,1;
%e A190015 6,8,1;
%e A190015 24,42,16,22,1;
%e A190015 120,264,180,192,136,52,1;
%e A190015 720,1920,1248,540,1824,2304,272,732,720,114,1;
%e A190015 5040,15840,10080,8064,18720,22752,9612,7056,10224,17928,3968,2538,3072,240,1;
%e A190015 40320,146160,92160,70560,32256,207360,249120,193536,73728,61560,144720,246816, 101844,142704,7936,51048,110448,34304,8334,11616,494,1;
%e A190015 Example for n=5:
%e A190015 partitions of number 9 into  5 parts in lexicographic order:
%e A190015 [1,1,1,1,5]
%e A190015 [1,1,1,2,4]
%e A190015 [1,1,1,3,3]
%e A190015 [1,1,2,2,3]
%e A190015 [1,2,2,2,2]
%e A190015 a(5) = (24*g(0)^4*g(4) +42*g(0)^3*g(1)*g(3) +16*g(0)^3*g(2)^2 +22*g(0)^2*g(1)^2*g(2) +g(0)*g(1)^4)/5!.
%o A190015 (Maxima)
%o A190015 /* array of triangle */
%o A190015 M:[1,1,2,1,6,8,1,24,42,16,22,1,120,264,180,192,136,52,1,720,1920,1248,540,1824,2304,272,732,720,114,1,5040,15840,10080,8064,18720,22752,9612,7056,10224,17928,3968,2538,3072,240,1,40320,146160,92160,70560,32256,207360,249120,193536,73728,61560,144720,246816,101844,142704,7936,51048,110448,34304,8334,11616,494,1];
%o A190015 /* function of triangle */
%o A190015 T(n,k):=M[sum(num_partitions(i),i,0,n-1)+k+1];
%o A190015 /* count number of partitions of n into m parts */
%o A190015 b(n,m):=if n<m then 0 else if m=1 then 1 else b(n-1,m-1)+b(n-m,m);
%o A190015 /* unranking partitions(n,m) , num - numbers partitions of lexicographic order */
%o A190015 array(pa,10);
%o A190015 gen_partitions(n,m,num,pos):= if n<m then return else
%o A190015                if m=1 then pa[pos]:n else
%o A190015                if num<b(n-1,m-1) then (pa[pos]:1, gen_partitions(n-1,m-1,num,pos+1)) else
%o A190015                if num<b(n-m,m)+b(n-1,m-1) then
%o A190015                 (gen_partitions(n-m,m,num-b(n-1,m-1),pos),
%o A190015                   for i:0 thru m-1 do pa[i+pos]:pa[i+pos]+1);
%o A190015 /* solve differential equation A'(x)=G(A(x)), G(x)=g(0)+g(1)*x+g(2)^x^2+...*/
%o A190015 /* gcoeff(n)=g(n) */
%o A190015 Solve(n,gcoeff):=block([s,h,num],s:0,for num:0 thru b(2*n-1,n)-1 do (
%o A190015 gen_partitions(2*n-1,n,num,0), s:s+T(n-1,num+1)*prod(gcoeff(pa[i]-1),i,0,n-1)),s/n!);
%o A190015 /*Test */
%o A190015 one(n):=1;
%o A190015 makelist(n!*Solve(n,one),n,1,9);
%o A190015 g(n):=2^n;
%o A190015 makelist(Solve(n,g),n,1,9);
%o A190015 (Maxima) /* Find triangle */
%o A190015 Co(n,k):=if k=1  then a(n) else sum(a(i+1)*Co(n-i-1,k-1),i,0,n-k);
%o A190015 a(n):=if n=1 then 1 else 1/n*sum(Co(n-1,k)*x(k),k,1,n-1);
%o A190015 makelist(ratsimp(n!*a(n)),n,1,5);
%o A190015 /* _Vladimir Kruchinin_, Jun 15 2012 */
%o A190015 (PARI) serlaplace( serreverse( intformal( 1 / sum(n=0, 9, eval(Str("g"n)) * x^n, x * O(x^9))))) /* _Michael Somos_, Oct 22 2014 */
%K A190015 nonn,tabf
%O A190015 0,3
%A A190015 _Vladimir Kruchinin_, May 04 2011