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A377104 E.g.f. A(x) satisfies [x^n] A(x)^A003057(n) = 0 for n >= 2, where A003057 is "n appears n-1 times.".

%I A377104 #10 Jul 23 2025 16:06:16
%S A377104 1,1,-1,4,-26,240,-2850,41160,-703640,13889400,-310575720,7752286080,
%T A377104 -213867376800,6462828372000,-212276818353600,7528584190327200,
%U A377104 -286677285603508800,11667274390189017600,-505448781205934966400,23223347244920039817600,-1127925105189437053699200,57737023232409594718444800
%N A377104 E.g.f. A(x) satisfies [x^n] A(x)^A003057(n) = 0 for n >= 2, where A003057 is "n appears n-1 times.".
%H A377104 Paul D. Hanna, <a href="/A377104/b377104.txt">Table of n, a(n) for n = 0..601</a>
%e A377104 E.g.f.: A(x) = 1 + x - x^2/2! + 4*x^3/3! - 26*x^4/4! + 240*x^5/5! - 2850*x^6/6! + 41160*x^7/7! - 703640*x^8/8! + 13889400*x^9/9! - 310575720*x^10/10! + 7752286080*x^11/11! - 213867376800*x^12/12! + 6462828372000*x^13/13! - 212276818353600*x^14/14! + 7528584190327200*x^15/15! + ...
%e A377104 RELATED TABLE.
%e A377104 The table of coefficients of x^k/k! in A(x)^n begins
%e A377104 n\k 0  1   2    3    4    5      6      7        8         9          10
%e A377104 1: [1, 1, -1,   4, -26, 240, -2850, 41160, -703640, 13889400, -310575720, ...];
%e A377104 2: [1, 2,  0,   2, -14, 140, -1720, 25060, -434280,  8662080, -194885040, ...];
%e A377104 3: [1, 3,  3,   0,   0,  30,  -480,  7560, -147000,  3136560,  -73364760, ...];
%e A377104 4: [1, 4,  8,   4,   4,   0,     0,     0,  -21560,   618240,  -16205280, ...];
%e A377104 5: [1, 5, 15,  20,  10,  20,    50, -1400,       0,        0,          0, 0, -1684359600, ...];
%e A377104 6: [1, 6, 24,  54,  54,  60,   120, -1260,  -11760,   -31920,    2000880, -65585520, 0, 0, 0, 0, 0, 24502922005161600, ...];
%e A377104 ...
%e A377104 in which there are (n-1) contiguous zeros in row n starting at k = (n-1)*(n-2)/2 + 2 for n >= 2.
%e A377104 Equivalently, [x^n] A(x)^A003057(n) = 0 for n >= 2, where
%e A377104 A003057 = [2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, ...] (offset 2).
%o A377104 (PARI) {A003057(n) = round( sqrt(2*(n-1)) ) + 1}
%o A377104 {a(n) = my(V=[1,1,0],A); for(i=0,n, V = concat(V,0); A = Ser(V); m = #V-2;
%o A377104 V[#V-1] = -polcoef(A^A003057(m), m)/A003057(m) ); n!*polcoef(A,n)}
%o A377104 for(n=1,30,print1(a(n),", "))
%Y A377104 Cf. A003057.
%K A377104 sign
%O A377104 0,4
%A A377104 _Paul D. Hanna_, Oct 31 2024