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

Author

Vladimir Kruchinin, May 04 2011

Keywords

• nonn
• tabf

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

Examples

			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!.
		

Programs

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
				

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 */