This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A377105 #16 Nov 01 2024 23:37:49
%S A377105 1,1,-1,3,-15,120,-1530,27630,-609570,15061950,-401747850,11409145650,
%T A377105 -343436893650,10953453262050,-370722834699750,13349217937623750,
%U A377105 -512802860608788750,21056741300679288750,-924609697617056519250,43346843603808212081250,-2162059111418145178443750
%N A377105 E.g.f. A(x) satisfies [x^k] A(x)^(n*(n+1)/2+1) = 0 for k = n*(n-1)/2+2 through k = n*(n+1)/2+1 for n >= 1, with a(0) = a(1) = 1.
%C A377105 Compare to: [x^k] 1/C(-x)^(n*(n+1)/2+2) = 0 for k = n*(n-1)/2+2 through k = n*(n+1)/2+1 for n >= 1 when C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers (A000108).
%e A377105 E.g.f.: A(x) = 1 + x - x^2/2! + 3*x^3/3! - 15*x^4/4! + 120*x^5/5! - 1530*x^6/6! + 27630*x^7/7! - 609570*x^8/8! + 15061950*x^9/9! - 401747850*x^10/10! + 11409145650*x^11/11! - 343436893650*x^12/12! + ...
%e A377105 RELATED TABLES.
%e A377105 The table of coefficients of x^k/k! in A(x)^n begins
%e A377105 n\k 0 1 2 3 4 5 6 7
%e A377105 1; [1, 1, -1, 3, -15, 120, -1530, 27630, ...];
%e A377105 2; [1, 2, 0, 0, 0, 30, -990, 25650, ...];
%e A377105 3; [1, 3, 3, -3, 9, 0, -900, 24300, ...];
%e A377105 4; [1, 4, 8, 0, 0, 60, -1260, 23580, ...];
%e A377105 5; [1, 5, 15, 15, -15, 120, -1350, 18450, ...];
%e A377105 6; [1, 6, 24, 48, 0, 90, -810, 8910, ...];
%e A377105 7; [1, 7, 35, 105, 105, 0, 0, 0, 11340, ...];
%e A377105 8; [1, 8, 48, 192, 384, 120, 360, -3240, 45000, ...];
%e A377105 9; [1, 9, 63, 315, 945, 1080, 270, -810, 18630, ...];
%e A377105 10; [1, 10, 80, 480, 1920, 3990, 2250, 2250, -8550, ...];
%e A377105 11; [1, 11, 99, 693, 3465, 10560, 13860, 5940, 0, 0, 0, 0, -1336500, ...];
%e A377105 ...
%e A377105 in which (at least) m contiguous zeros are found in row n = m*(m+1)/2+1 starting at k = m*(m-1)/2+2 through k = n for m >= 1.
%e A377105 Explicitly, zeros are found in the above table at (row,column) = (n,k) given by
%e A377105 [(2,2), (4,3), (4,4), (7,5), (7,6), (7,7), (11,8), (11,9), (11,10), (11,11), (16,12), (16,13), (16,14), (16,15), (16,16), ...]
%e A377105 and by definition this pattern of zeros continues indefinitely.
%e A377105 RELATED SERIES.
%e A377105 Let B(x) = A(x/B(x)) and A(x) = B(x*A(x)) then
%e A377105 B(x) = 1 + x + x^2/2! + 0*x^3/3! - 3*x^4/4! + 15*x^5/5! + 0*x^6/6! - 405*x^7/7! + 2070*x^8/8! + 1620*x^9/9! + 0*x^10/10! - 1935900*x^11/11! + 27592650*x^12/12! - 63022050*x^13/13! - 1958737950*x^14/14! + 0*x^15/15! + ... + A377106(n)*x^n/n! + ...
%e A377105 where [x^(n*(n+1)/2)] B(x) = 0 for n >= 2.
%e A377105 1/A(-x) = 1 + x + 3*x^2/2! + 15*x^3/3! + 105*x^4/4! + 960*x^5/5! + 11160*x^6/6! + 163080*x^7/7! + 2948580*x^8/8! + 64082340*x^9/9! + 1614532500*x^10/10! + 45628134300*x^11/11! + 1411847739000*x^12/12! + ...
%o A377105 (PARI) {b(m) = my(t = (sqrtint(8*m-8)+1)\2); t*(t+1)/2+1}
%o A377105 {a(n) = my(V=[1, 1, 0], A); for(i=0, n, V = concat(V, 0); A = Ser(V); m = #V-2;
%o A377105 V[#V-1] = -polcoef(A^b(m), m)/b(m) ); n!*polcoef(A, n)}
%o A377105 for(n=1, 30, print1(a(n), ", "))
%Y A377105 Cf. A377106.
%K A377105 sign
%O A377105 0,4
%A A377105 _Paul D. Hanna_, Nov 01 2024