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 A190015 Triangle T(n,k) for solving differential equation A'(x)=G(A(x)), G(0)!=0. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified September 23 12:06 EDT 2021. Contains 347616 sequences. (Running on oeis4.)