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

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

Offset

0,3

Comments

For solving the differential equation A'(x)=G(A(x)), where G(0)!=0,

a(n) = 1/n!*sum(pi(i) in P(2*n-1,n), T(n,i)*prod(j=1..n, g(k_j-1))),

where pi(i) is the partition of 2*n-1 into n parts in lexicographic order P(2*n-1,n).

G(x) = g(0)+g(1)*x+g(2)*x^2+...

Examples

A003422 A'(x)=A(x)+1/(1-x)

A000108 A'(x)=1/(1-2*A(x)),

A001147 A'(x)=1/(1-A(x))

A007489 A'(x)=A(x)+x/(1-x)^2+1.

A006351 B'(x)=(1+B(x))/(1-B(x))

A029768 A'(x)=log(1/(1-A(x)))+1.

A001662 B'(x)=1/(1+B(x))

A180254 A'(x)=(1-sqrt(1-4*A(x)))/2

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

Links

Table of n, a(n) for n=0..66.

Example

Triangle begins:
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;
Example for n=5:
partitions of number 9 into  5 parts in lexicographic order:
[1,1,1,1,5]
[1,1,1,2,4]
[1,1,1,3,3]
[1,1,2,2,3]
[1,2,2,2,2]
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!.

Prog

(Maxima)
/* array of triangle */
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];
/* function of triangle */
T(n, k):=M[sum(num_partitions(i), i, 0, n-1)+k+1];
/* count number of partitions of n into m parts */
b(n, m):=if n<m then 0 else if m=1 then 1 else b(n-1, m-1)+b(n-m, m);
/* unranking partitions(n, m) , num - numbers partitions of lexicographic order */
array(pa, 10);
gen_partitions(n, m, num, pos):= if n<m then return else
               if m=1 then pa[pos]:n else
               if num<b(n-1, m-1) then (pa[pos]:1, gen_partitions(n-1, m-1, num, pos+1)) else
               if num<b(n-m, m)+b(n-1, m-1) then
                (gen_partitions(n-m, m, num-b(n-1, m-1), pos),
                  for i:0 thru m-1 do pa[i+pos]:pa[i+pos]+1);
/* solve differential equation A'(x)=G(A(x)), G(x)=g(0)+g(1)*x+g(2)^x^2+...*/
/* gcoeff(n)=g(n) */
Solve(n, gcoeff):=block([s, h, num], s:0, for num:0 thru b(2*n-1, n)-1 do (
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!);
/*Test */
one(n):=1;
makelist(n!*Solve(n, one), n, 1, 9);
g(n):=2^n;
makelist(Solve(n, g), n, 1, 9);
(Maxima) /* Find triangle */
Co(n, k):=if k=1  then a(n) else sum(a(i+1)*Co(n-i-1, k-1), i, 0, n-k);
a(n):=if n=1 then 1 else 1/n*sum(Co(n-1, k)*x(k), k, 1, n-1);
makelist(ratsimp(n!*a(n)), n, 1, 5);
/* Vladimir Kruchinin, Jun 15 2012 */
(PARI) serlaplace( serreverse( intformal( 1 / sum(n=0, 9, eval(Str("g"n)) * x^n, x * O(x^9))))) /* Michael Somos, Oct 22 2014 */

Crossrefs

Sequence in context: A319511 A110608 A318397 * A112007 A113374 A136470

Adjacent sequences: A190012 A190013 A190014 * A190016 A190017 A190018

Keyword

nonn,tabf

Author

Vladimir Kruchinin, May 04 2011

Status

approved