A317338 O.g.f. A(x) satisfies: [x^n] exp( n*x*A(x) ) * (n+1 - n*A(x)) = 0 for n >= 1.

1, 1, 0, -3, -5, 10, 58, 23, -557, -1421, 4094, 28316, -52, -449150, -970286, 5908939, 31046627, -49583353, -750617284, -544416915, 15819383275, 46795708732, -288245326872, -1808819140124, 3784215933076, 57664747490276, 14416027504376, -1664155475303224, -3937904190952656, 43893853942734810, 219165998056699650

Offset: 0

Author

Paul D. Hanna, Aug 01 2018

Keywords

• sign

Comments

Compare: the factorial series F(x) = Sum_{n>=0} n!*x^n satisfies

(1) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n >= 1,

(2) [x^n] exp( n*x*F(x) ) * (2 - F(x)) = 0 for n >= 1,

(3) [x^n] exp( n^2*x*F(x) ) * (n + 1 - n*F(x)) = 0 for n >= 1.

It is remarkable that this sequence should consist entirely of integers.

Examples

			O.g.f.: A(x) = 1 + x - 3*x^3 - 5*x^4 + 10*x^5 + 58*x^6 + 23*x^7 - 557*x^8 - 1421*x^9 + 4094*x^10 + 28316*x^11 - 52*x^12 - 449150*x^13 - 970286*x^14 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n*x*A(x) ) * (n+1 - n*A(x)) begins:
n=1: [1, 0, 1, 16, 117, -704, -35075, -200304, 17660041, ...];
n=2: [1, 0, 0, 20, 288, 912, -51200, -888480, 19165440, ...];
n=3: [1, 0, -3, 0, 333, 3888, -27135, -1471824, 4665465, ...];
n=4: [1, 0, -8, -56, 0, 5344, 33280, -1317312, -15647744, ...];
n=5: [1, 0, -15, -160, -1035, 0, 81325, -180000, -25008375, ...];
n=6: [1, 0, -24, -324, -3168, -20304, 0, 1156896, -10209024, ...];
n=7: [1, 0, -35, -560, -6867, -67088, -422975, 0, 19205305, ...];
n=8: [1, 0, -48, -880, -12672, -155712, -1525760, -9408384, 0, ...];
n=9: [1, 0, -63, -1296, -21195, -305856, -3806595, -37346832, -230393079, 0, ...]; ...
in which the coefficients of x^n in row n form a diagonal of zeros.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A317337, A305110, A305114, A305115, A305116.

Programs

PARI

{a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*x*Ser(A) ) * (m - (m-1)*Ser(A)) )[m]/(m-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))