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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
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
Sequence in context: A368221 A077632 A347976 * A361640 A323506 A357988
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.)