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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, -128, 576, -720, -320, 216, 576, 160, -108, -96, -180, -72, 40, 36, 28, -8 (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 Table of n, a(n) for n=0..44. 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; in the reversed order) 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 MATHEMATICA rows[n_] := {{1}}~Join~With[{s = 1/(1 + Sum[(k+1) u[k] x^k, {k, n}] + O[x]^(n+1))}, Table[Coefficient[s, x^k Product[u[t], {t, p}]], {k, n}, {p, Reverse@Sort[Sort /@ IntegerPartitions[k]]}]]; rows[7] // Flatten (* Andrey Zabolotskiy, Feb 19 2024 *) PROG (PARI) C(v)={my(S=Set(v)); (-1)^(#v)*(#v)!*prod(i=1, #S, my(x=S[i], e=#select(y-> y==x, v)); (x+1)^e/e! )} row(n)=[C(Vec(p)) | p<-Vecrev(partitions(n))] { for(n=0, 7, print(row(n))) } \\ Andrew Howroyd, Feb 19 2024 CROSSREFS Cf. A181289, A133437, A145271, A133314, A003480. Sequence in context: A368221 A077632 A347976 * A361640 A323506 A357988 Adjacent sequences: A253719 A253720 A253721 * A253723 A253724 A253725 KEYWORD sign,tabf AUTHOR Tom Copeland, May 02 2015 EXTENSIONS Row 7 added by Andrey Zabolotskiy, Feb 19 2024 STATUS approved

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Last modified June 15 06:51 EDT 2024. Contains 373402 sequences. (Running on oeis4.)