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 A253722 Triangle read by rows: coefficients of the partition polynomials for the reciprocal of the derivative of a power series, g(x)= 1/h'(x). 1
 1, -2, 4, -3, -8, 12, -4, 16, -36, 9, 16, -5, -32, 96, -54, -48, 24, 20, -6, 64, -240, 216, 128, -27, -144, -60, 16, 30, 24, -7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This entry contains the integer coefficients of the partition polynomials P(n;h_1,h_2,...,h_(n+1)) for the reciprocal g(x) of the derivative of a power series in terms of the coefficients of the power series; i.e., g(x) = 1/[dh(x)/dx] = 1/[h_1 + 2*h_2 * x + 3*h_3 * x^2 +  ...] = sum[n>=0, (h_1)^(-(n+1)) * P(n;h_1,...,h_(n+1)) * x^n]. This is a signed refinement of reversed A181289. See A145271, A133437, and A133314 for relations to compositional and multiplicative inversions. LINKS M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. FORMULA For the partition (1')^e(1)*(2')^e(2)*...*(n')^e(n) in P(m;...), the unsigned integer coefficient is [e(2)+e(3)+...+e(n)]! * [2^e(2)*3^e(3)*...*n^e(n)]/[e(2)!*e(3)!*...*e(n)!] with the sign determined by (-1)^[e(1) + m]. The partitions of P(m;..) are formed by adding one to each index of the partitions of m of Abramowitz and Stegun's partition table (p. 831) and appending (1')^e(1) as a factor to obtain a partition of 2m. Row sums are 1,-2,1,0,0,0,... . Row sums of the unsigned coefficients are A003480. EXAMPLE Let h(x) = h_0 + h_1 * x + h_2 * x^2 + ... . Then g(x) = 1/h'(x) = 1/[h_1 + 2*h_2 * x + 3*h_3 * x^2 + ...] = (h_1)^(-1) P(0;h_1) + (h_1)^(-2) * P(1;h_1,h_2) * x + (h_1)^(-3) * P(2;h_1,h_2,h_3) * x^2 + ... , and, with h_n = (n'), the first few partition polynomials are P(0;..)=  1 P(1;..)= -2 (2') P(2;..)=  4 (2')^2 - 3 (3')(1') P(3;..)= -8 (2')^3 + 12 (3')(2')(1') - 4 (4')(1')^2 P(4;..)= 16 (2')^4 - 36 (2')^2(3')(1') + [9 (3')^2 + 16 (4')(2')](1')^2 - 5 (5')(1')^3 P(5;..)= -32 (2')^5 + 96 (2')^3(3')(1') + [-54 (3')^2(2') - 48 (4')(2')^2](1')^2 + [24 (3')(4') + 20 (5')(2')](1')^3 - 6 (6')(1')^4 P(6;..)= 64 (2')^6 - 240 (2')^4(3')(1') + [216 (3')^2(2') + 128 (4')(2')^3](1')^2 - [27 (3')^3 + 144 (4')(3')(2') + 60 (5')(2')^2](1')^3 + [16 (4')^2 + 30 (5')(3') + 24 (6')(2')](1')^4 - 7 (7')(1')^5 CROSSREFS Cf. A181289, A133437, A145271, A133314, A003480. Sequence in context: A186003 A077624 A077632 * A323506 A302747 A193949 Adjacent sequences:  A253719 A253720 A253721 * A253723 A253724 A253725 KEYWORD sign,tabf AUTHOR Tom Copeland, May 02 2015 STATUS approved

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Last modified April 20 01:55 EDT 2021. Contains 343118 sequences. (Running on oeis4.)